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Probability II. Denoted by P(Event). Probability. This method for calculating probabilities is only appropriate when the outcomes of the sample space are equally likely. The relative frequency at which a chance experiment occurs Flip a fair coin 30 times & get 17 heads. - PowerPoint PPT Presentation
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Sample space
bull the collection of all possible outcomes of a chance experimentndashRoll a die
S=123456
Event bull any collection of outcomes
from the sample space
ndashRolling a prime E= 235
Complement
bull Consists of all outcomes that are not in the event
ndashNot rolling an even
EC=135
Union bull the event A or B happening
bull consists of all outcomes that are in at least one of the two eventsndash Rolling a prime or even number
E=23456
BAE
Intersection bull the event A and B happening
bull consists of all outcomes that are in both eventsndash Drawing a red card and a ldquo2rdquo E = 2
hearts 2 diamonds
BAE
Mutually Exclusive (disjoint)
bull two events have no outcomes in common
ndashRoll a ldquo2rdquo and a ldquo5rdquo
Probabilitybull Denoted by P(Event)
outcomes total
outcomes favorable)( EP
This method for calculating probabilities is only appropriate when the outcomes of the sample space are equally likely
Experimental Probability
bull The relative frequency at which a chance experiment occursndashFlip a fair coin 30 times amp get 17
heads
30
17
Law of Large Numbers
bull As the number of repetitions of a chance experiment increase the difference between the relative frequency of occurrence for an event and the true probability approaches zero
Basic Rules of ProbabilityRule 1 Legitimate Values
For any event E 0 lt P(E) lt 1
Rule 2 Sample spaceIf S is the sample space P(S) = 1
Rule 3 Complement
For any event E P(E) + P(not E) = 1
Complement of an eventhellip
Ec
Independentbull Two events are independent if knowing that
one will occur (or has occurred) does not change the probability that the other occurs
So Chuck what are you going to do next
year after graduation
Oh I plan on going to a culinary school to become a chef ndash I
just love to cook
Teacher to student in school
Teacher to student in front of parent
Oh remember ndash Irsquom going to college to
become an engineer
Are Chuckrsquos responses independent
Rule 4 Multiplication
If two events A amp B are independent
General rule
P(B) P(A) B) ampP(A
A)|P(B P(A) B) ampP(A
)( BAP
Independent)( BAP
Yes
)()( BPAP
What does this mean
Ex 1) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that both bulbs are defective
Can you assume they are independent
00250505D) amp P(D
)( BAP
Independent)( BAP
Yes No
)()( BPAP )|()( ABPAP
Ex 2) There are seven girls and eight boys in a math class The teacher selects two students at random to answer questions on the board What is the probability that both students are girls
Are these events independent
2146
157
G) amp P(G
Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades
Are the cards independent NO
P(A amp B) = P(A) P(B|A)
P(Spade amp Spade) = 14 1251 = 117
The probability of getting a spade given that a spade has already been drawn
Rule 5 AdditionIf two events E amp F are disjoint
P(E or F) = P(E) + P(F)
(General) If two events E amp F are not disjoint
P(E or F) = P(E) + P(F) ndash P(E amp F)
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota (Note these are not simple events since there are many types of each brand) Suppose that P(H) = 25 P(N) = 18 P(T) = 14
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAILS Roll is LESS THAN 3
P (tails or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
Rule 6 At least one
The probability that at least one outcome happens is 1 minus the probability that no outcomes happen
P(at least 1) = 1 ndash P(none)
Ex 8 revisited) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store
What is the probability that at least one bulb is defective
P(at least one) = 1 - P(DC amp DC)
0975
Ex 9) For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something
P(at least one winning symbol) =
1 ndash P(no winning symbols) 1 - 96 = 4686
Rule 7 Conditional Probability
bull A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
Event bull any collection of outcomes
from the sample space
ndashRolling a prime E= 235
Complement
bull Consists of all outcomes that are not in the event
ndashNot rolling an even
EC=135
Union bull the event A or B happening
bull consists of all outcomes that are in at least one of the two eventsndash Rolling a prime or even number
E=23456
BAE
Intersection bull the event A and B happening
bull consists of all outcomes that are in both eventsndash Drawing a red card and a ldquo2rdquo E = 2
hearts 2 diamonds
BAE
Mutually Exclusive (disjoint)
bull two events have no outcomes in common
ndashRoll a ldquo2rdquo and a ldquo5rdquo
Probabilitybull Denoted by P(Event)
outcomes total
outcomes favorable)( EP
This method for calculating probabilities is only appropriate when the outcomes of the sample space are equally likely
Experimental Probability
bull The relative frequency at which a chance experiment occursndashFlip a fair coin 30 times amp get 17
heads
30
17
Law of Large Numbers
bull As the number of repetitions of a chance experiment increase the difference between the relative frequency of occurrence for an event and the true probability approaches zero
Basic Rules of ProbabilityRule 1 Legitimate Values
For any event E 0 lt P(E) lt 1
Rule 2 Sample spaceIf S is the sample space P(S) = 1
Rule 3 Complement
For any event E P(E) + P(not E) = 1
Complement of an eventhellip
Ec
Independentbull Two events are independent if knowing that
one will occur (or has occurred) does not change the probability that the other occurs
So Chuck what are you going to do next
year after graduation
Oh I plan on going to a culinary school to become a chef ndash I
just love to cook
Teacher to student in school
Teacher to student in front of parent
Oh remember ndash Irsquom going to college to
become an engineer
Are Chuckrsquos responses independent
Rule 4 Multiplication
If two events A amp B are independent
General rule
P(B) P(A) B) ampP(A
A)|P(B P(A) B) ampP(A
)( BAP
Independent)( BAP
Yes
)()( BPAP
What does this mean
Ex 1) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that both bulbs are defective
Can you assume they are independent
00250505D) amp P(D
)( BAP
Independent)( BAP
Yes No
)()( BPAP )|()( ABPAP
Ex 2) There are seven girls and eight boys in a math class The teacher selects two students at random to answer questions on the board What is the probability that both students are girls
Are these events independent
2146
157
G) amp P(G
Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades
Are the cards independent NO
P(A amp B) = P(A) P(B|A)
P(Spade amp Spade) = 14 1251 = 117
The probability of getting a spade given that a spade has already been drawn
Rule 5 AdditionIf two events E amp F are disjoint
P(E or F) = P(E) + P(F)
(General) If two events E amp F are not disjoint
P(E or F) = P(E) + P(F) ndash P(E amp F)
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota (Note these are not simple events since there are many types of each brand) Suppose that P(H) = 25 P(N) = 18 P(T) = 14
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAILS Roll is LESS THAN 3
P (tails or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
Rule 6 At least one
The probability that at least one outcome happens is 1 minus the probability that no outcomes happen
P(at least 1) = 1 ndash P(none)
Ex 8 revisited) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store
What is the probability that at least one bulb is defective
P(at least one) = 1 - P(DC amp DC)
0975
Ex 9) For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something
P(at least one winning symbol) =
1 ndash P(no winning symbols) 1 - 96 = 4686
Rule 7 Conditional Probability
bull A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
Complement
bull Consists of all outcomes that are not in the event
ndashNot rolling an even
EC=135
Union bull the event A or B happening
bull consists of all outcomes that are in at least one of the two eventsndash Rolling a prime or even number
E=23456
BAE
Intersection bull the event A and B happening
bull consists of all outcomes that are in both eventsndash Drawing a red card and a ldquo2rdquo E = 2
hearts 2 diamonds
BAE
Mutually Exclusive (disjoint)
bull two events have no outcomes in common
ndashRoll a ldquo2rdquo and a ldquo5rdquo
Probabilitybull Denoted by P(Event)
outcomes total
outcomes favorable)( EP
This method for calculating probabilities is only appropriate when the outcomes of the sample space are equally likely
Experimental Probability
bull The relative frequency at which a chance experiment occursndashFlip a fair coin 30 times amp get 17
heads
30
17
Law of Large Numbers
bull As the number of repetitions of a chance experiment increase the difference between the relative frequency of occurrence for an event and the true probability approaches zero
Basic Rules of ProbabilityRule 1 Legitimate Values
For any event E 0 lt P(E) lt 1
Rule 2 Sample spaceIf S is the sample space P(S) = 1
Rule 3 Complement
For any event E P(E) + P(not E) = 1
Complement of an eventhellip
Ec
Independentbull Two events are independent if knowing that
one will occur (or has occurred) does not change the probability that the other occurs
So Chuck what are you going to do next
year after graduation
Oh I plan on going to a culinary school to become a chef ndash I
just love to cook
Teacher to student in school
Teacher to student in front of parent
Oh remember ndash Irsquom going to college to
become an engineer
Are Chuckrsquos responses independent
Rule 4 Multiplication
If two events A amp B are independent
General rule
P(B) P(A) B) ampP(A
A)|P(B P(A) B) ampP(A
)( BAP
Independent)( BAP
Yes
)()( BPAP
What does this mean
Ex 1) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that both bulbs are defective
Can you assume they are independent
00250505D) amp P(D
)( BAP
Independent)( BAP
Yes No
)()( BPAP )|()( ABPAP
Ex 2) There are seven girls and eight boys in a math class The teacher selects two students at random to answer questions on the board What is the probability that both students are girls
Are these events independent
2146
157
G) amp P(G
Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades
Are the cards independent NO
P(A amp B) = P(A) P(B|A)
P(Spade amp Spade) = 14 1251 = 117
The probability of getting a spade given that a spade has already been drawn
Rule 5 AdditionIf two events E amp F are disjoint
P(E or F) = P(E) + P(F)
(General) If two events E amp F are not disjoint
P(E or F) = P(E) + P(F) ndash P(E amp F)
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota (Note these are not simple events since there are many types of each brand) Suppose that P(H) = 25 P(N) = 18 P(T) = 14
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAILS Roll is LESS THAN 3
P (tails or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
Rule 6 At least one
The probability that at least one outcome happens is 1 minus the probability that no outcomes happen
P(at least 1) = 1 ndash P(none)
Ex 8 revisited) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store
What is the probability that at least one bulb is defective
P(at least one) = 1 - P(DC amp DC)
0975
Ex 9) For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something
P(at least one winning symbol) =
1 ndash P(no winning symbols) 1 - 96 = 4686
Rule 7 Conditional Probability
bull A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
Union bull the event A or B happening
bull consists of all outcomes that are in at least one of the two eventsndash Rolling a prime or even number
E=23456
BAE
Intersection bull the event A and B happening
bull consists of all outcomes that are in both eventsndash Drawing a red card and a ldquo2rdquo E = 2
hearts 2 diamonds
BAE
Mutually Exclusive (disjoint)
bull two events have no outcomes in common
ndashRoll a ldquo2rdquo and a ldquo5rdquo
Probabilitybull Denoted by P(Event)
outcomes total
outcomes favorable)( EP
This method for calculating probabilities is only appropriate when the outcomes of the sample space are equally likely
Experimental Probability
bull The relative frequency at which a chance experiment occursndashFlip a fair coin 30 times amp get 17
heads
30
17
Law of Large Numbers
bull As the number of repetitions of a chance experiment increase the difference between the relative frequency of occurrence for an event and the true probability approaches zero
Basic Rules of ProbabilityRule 1 Legitimate Values
For any event E 0 lt P(E) lt 1
Rule 2 Sample spaceIf S is the sample space P(S) = 1
Rule 3 Complement
For any event E P(E) + P(not E) = 1
Complement of an eventhellip
Ec
Independentbull Two events are independent if knowing that
one will occur (or has occurred) does not change the probability that the other occurs
So Chuck what are you going to do next
year after graduation
Oh I plan on going to a culinary school to become a chef ndash I
just love to cook
Teacher to student in school
Teacher to student in front of parent
Oh remember ndash Irsquom going to college to
become an engineer
Are Chuckrsquos responses independent
Rule 4 Multiplication
If two events A amp B are independent
General rule
P(B) P(A) B) ampP(A
A)|P(B P(A) B) ampP(A
)( BAP
Independent)( BAP
Yes
)()( BPAP
What does this mean
Ex 1) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that both bulbs are defective
Can you assume they are independent
00250505D) amp P(D
)( BAP
Independent)( BAP
Yes No
)()( BPAP )|()( ABPAP
Ex 2) There are seven girls and eight boys in a math class The teacher selects two students at random to answer questions on the board What is the probability that both students are girls
Are these events independent
2146
157
G) amp P(G
Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades
Are the cards independent NO
P(A amp B) = P(A) P(B|A)
P(Spade amp Spade) = 14 1251 = 117
The probability of getting a spade given that a spade has already been drawn
Rule 5 AdditionIf two events E amp F are disjoint
P(E or F) = P(E) + P(F)
(General) If two events E amp F are not disjoint
P(E or F) = P(E) + P(F) ndash P(E amp F)
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota (Note these are not simple events since there are many types of each brand) Suppose that P(H) = 25 P(N) = 18 P(T) = 14
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAILS Roll is LESS THAN 3
P (tails or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
Rule 6 At least one
The probability that at least one outcome happens is 1 minus the probability that no outcomes happen
P(at least 1) = 1 ndash P(none)
Ex 8 revisited) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store
What is the probability that at least one bulb is defective
P(at least one) = 1 - P(DC amp DC)
0975
Ex 9) For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something
P(at least one winning symbol) =
1 ndash P(no winning symbols) 1 - 96 = 4686
Rule 7 Conditional Probability
bull A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
Intersection bull the event A and B happening
bull consists of all outcomes that are in both eventsndash Drawing a red card and a ldquo2rdquo E = 2
hearts 2 diamonds
BAE
Mutually Exclusive (disjoint)
bull two events have no outcomes in common
ndashRoll a ldquo2rdquo and a ldquo5rdquo
Probabilitybull Denoted by P(Event)
outcomes total
outcomes favorable)( EP
This method for calculating probabilities is only appropriate when the outcomes of the sample space are equally likely
Experimental Probability
bull The relative frequency at which a chance experiment occursndashFlip a fair coin 30 times amp get 17
heads
30
17
Law of Large Numbers
bull As the number of repetitions of a chance experiment increase the difference between the relative frequency of occurrence for an event and the true probability approaches zero
Basic Rules of ProbabilityRule 1 Legitimate Values
For any event E 0 lt P(E) lt 1
Rule 2 Sample spaceIf S is the sample space P(S) = 1
Rule 3 Complement
For any event E P(E) + P(not E) = 1
Complement of an eventhellip
Ec
Independentbull Two events are independent if knowing that
one will occur (or has occurred) does not change the probability that the other occurs
So Chuck what are you going to do next
year after graduation
Oh I plan on going to a culinary school to become a chef ndash I
just love to cook
Teacher to student in school
Teacher to student in front of parent
Oh remember ndash Irsquom going to college to
become an engineer
Are Chuckrsquos responses independent
Rule 4 Multiplication
If two events A amp B are independent
General rule
P(B) P(A) B) ampP(A
A)|P(B P(A) B) ampP(A
)( BAP
Independent)( BAP
Yes
)()( BPAP
What does this mean
Ex 1) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that both bulbs are defective
Can you assume they are independent
00250505D) amp P(D
)( BAP
Independent)( BAP
Yes No
)()( BPAP )|()( ABPAP
Ex 2) There are seven girls and eight boys in a math class The teacher selects two students at random to answer questions on the board What is the probability that both students are girls
Are these events independent
2146
157
G) amp P(G
Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades
Are the cards independent NO
P(A amp B) = P(A) P(B|A)
P(Spade amp Spade) = 14 1251 = 117
The probability of getting a spade given that a spade has already been drawn
Rule 5 AdditionIf two events E amp F are disjoint
P(E or F) = P(E) + P(F)
(General) If two events E amp F are not disjoint
P(E or F) = P(E) + P(F) ndash P(E amp F)
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota (Note these are not simple events since there are many types of each brand) Suppose that P(H) = 25 P(N) = 18 P(T) = 14
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAILS Roll is LESS THAN 3
P (tails or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
Rule 6 At least one
The probability that at least one outcome happens is 1 minus the probability that no outcomes happen
P(at least 1) = 1 ndash P(none)
Ex 8 revisited) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store
What is the probability that at least one bulb is defective
P(at least one) = 1 - P(DC amp DC)
0975
Ex 9) For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something
P(at least one winning symbol) =
1 ndash P(no winning symbols) 1 - 96 = 4686
Rule 7 Conditional Probability
bull A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
Mutually Exclusive (disjoint)
bull two events have no outcomes in common
ndashRoll a ldquo2rdquo and a ldquo5rdquo
Probabilitybull Denoted by P(Event)
outcomes total
outcomes favorable)( EP
This method for calculating probabilities is only appropriate when the outcomes of the sample space are equally likely
Experimental Probability
bull The relative frequency at which a chance experiment occursndashFlip a fair coin 30 times amp get 17
heads
30
17
Law of Large Numbers
bull As the number of repetitions of a chance experiment increase the difference between the relative frequency of occurrence for an event and the true probability approaches zero
Basic Rules of ProbabilityRule 1 Legitimate Values
For any event E 0 lt P(E) lt 1
Rule 2 Sample spaceIf S is the sample space P(S) = 1
Rule 3 Complement
For any event E P(E) + P(not E) = 1
Complement of an eventhellip
Ec
Independentbull Two events are independent if knowing that
one will occur (or has occurred) does not change the probability that the other occurs
So Chuck what are you going to do next
year after graduation
Oh I plan on going to a culinary school to become a chef ndash I
just love to cook
Teacher to student in school
Teacher to student in front of parent
Oh remember ndash Irsquom going to college to
become an engineer
Are Chuckrsquos responses independent
Rule 4 Multiplication
If two events A amp B are independent
General rule
P(B) P(A) B) ampP(A
A)|P(B P(A) B) ampP(A
)( BAP
Independent)( BAP
Yes
)()( BPAP
What does this mean
Ex 1) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that both bulbs are defective
Can you assume they are independent
00250505D) amp P(D
)( BAP
Independent)( BAP
Yes No
)()( BPAP )|()( ABPAP
Ex 2) There are seven girls and eight boys in a math class The teacher selects two students at random to answer questions on the board What is the probability that both students are girls
Are these events independent
2146
157
G) amp P(G
Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades
Are the cards independent NO
P(A amp B) = P(A) P(B|A)
P(Spade amp Spade) = 14 1251 = 117
The probability of getting a spade given that a spade has already been drawn
Rule 5 AdditionIf two events E amp F are disjoint
P(E or F) = P(E) + P(F)
(General) If two events E amp F are not disjoint
P(E or F) = P(E) + P(F) ndash P(E amp F)
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota (Note these are not simple events since there are many types of each brand) Suppose that P(H) = 25 P(N) = 18 P(T) = 14
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAILS Roll is LESS THAN 3
P (tails or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
Rule 6 At least one
The probability that at least one outcome happens is 1 minus the probability that no outcomes happen
P(at least 1) = 1 ndash P(none)
Ex 8 revisited) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store
What is the probability that at least one bulb is defective
P(at least one) = 1 - P(DC amp DC)
0975
Ex 9) For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something
P(at least one winning symbol) =
1 ndash P(no winning symbols) 1 - 96 = 4686
Rule 7 Conditional Probability
bull A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
Probabilitybull Denoted by P(Event)
outcomes total
outcomes favorable)( EP
This method for calculating probabilities is only appropriate when the outcomes of the sample space are equally likely
Experimental Probability
bull The relative frequency at which a chance experiment occursndashFlip a fair coin 30 times amp get 17
heads
30
17
Law of Large Numbers
bull As the number of repetitions of a chance experiment increase the difference between the relative frequency of occurrence for an event and the true probability approaches zero
Basic Rules of ProbabilityRule 1 Legitimate Values
For any event E 0 lt P(E) lt 1
Rule 2 Sample spaceIf S is the sample space P(S) = 1
Rule 3 Complement
For any event E P(E) + P(not E) = 1
Complement of an eventhellip
Ec
Independentbull Two events are independent if knowing that
one will occur (or has occurred) does not change the probability that the other occurs
So Chuck what are you going to do next
year after graduation
Oh I plan on going to a culinary school to become a chef ndash I
just love to cook
Teacher to student in school
Teacher to student in front of parent
Oh remember ndash Irsquom going to college to
become an engineer
Are Chuckrsquos responses independent
Rule 4 Multiplication
If two events A amp B are independent
General rule
P(B) P(A) B) ampP(A
A)|P(B P(A) B) ampP(A
)( BAP
Independent)( BAP
Yes
)()( BPAP
What does this mean
Ex 1) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that both bulbs are defective
Can you assume they are independent
00250505D) amp P(D
)( BAP
Independent)( BAP
Yes No
)()( BPAP )|()( ABPAP
Ex 2) There are seven girls and eight boys in a math class The teacher selects two students at random to answer questions on the board What is the probability that both students are girls
Are these events independent
2146
157
G) amp P(G
Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades
Are the cards independent NO
P(A amp B) = P(A) P(B|A)
P(Spade amp Spade) = 14 1251 = 117
The probability of getting a spade given that a spade has already been drawn
Rule 5 AdditionIf two events E amp F are disjoint
P(E or F) = P(E) + P(F)
(General) If two events E amp F are not disjoint
P(E or F) = P(E) + P(F) ndash P(E amp F)
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota (Note these are not simple events since there are many types of each brand) Suppose that P(H) = 25 P(N) = 18 P(T) = 14
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAILS Roll is LESS THAN 3
P (tails or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
Rule 6 At least one
The probability that at least one outcome happens is 1 minus the probability that no outcomes happen
P(at least 1) = 1 ndash P(none)
Ex 8 revisited) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store
What is the probability that at least one bulb is defective
P(at least one) = 1 - P(DC amp DC)
0975
Ex 9) For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something
P(at least one winning symbol) =
1 ndash P(no winning symbols) 1 - 96 = 4686
Rule 7 Conditional Probability
bull A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
Experimental Probability
bull The relative frequency at which a chance experiment occursndashFlip a fair coin 30 times amp get 17
heads
30
17
Law of Large Numbers
bull As the number of repetitions of a chance experiment increase the difference between the relative frequency of occurrence for an event and the true probability approaches zero
Basic Rules of ProbabilityRule 1 Legitimate Values
For any event E 0 lt P(E) lt 1
Rule 2 Sample spaceIf S is the sample space P(S) = 1
Rule 3 Complement
For any event E P(E) + P(not E) = 1
Complement of an eventhellip
Ec
Independentbull Two events are independent if knowing that
one will occur (or has occurred) does not change the probability that the other occurs
So Chuck what are you going to do next
year after graduation
Oh I plan on going to a culinary school to become a chef ndash I
just love to cook
Teacher to student in school
Teacher to student in front of parent
Oh remember ndash Irsquom going to college to
become an engineer
Are Chuckrsquos responses independent
Rule 4 Multiplication
If two events A amp B are independent
General rule
P(B) P(A) B) ampP(A
A)|P(B P(A) B) ampP(A
)( BAP
Independent)( BAP
Yes
)()( BPAP
What does this mean
Ex 1) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that both bulbs are defective
Can you assume they are independent
00250505D) amp P(D
)( BAP
Independent)( BAP
Yes No
)()( BPAP )|()( ABPAP
Ex 2) There are seven girls and eight boys in a math class The teacher selects two students at random to answer questions on the board What is the probability that both students are girls
Are these events independent
2146
157
G) amp P(G
Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades
Are the cards independent NO
P(A amp B) = P(A) P(B|A)
P(Spade amp Spade) = 14 1251 = 117
The probability of getting a spade given that a spade has already been drawn
Rule 5 AdditionIf two events E amp F are disjoint
P(E or F) = P(E) + P(F)
(General) If two events E amp F are not disjoint
P(E or F) = P(E) + P(F) ndash P(E amp F)
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota (Note these are not simple events since there are many types of each brand) Suppose that P(H) = 25 P(N) = 18 P(T) = 14
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAILS Roll is LESS THAN 3
P (tails or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
Rule 6 At least one
The probability that at least one outcome happens is 1 minus the probability that no outcomes happen
P(at least 1) = 1 ndash P(none)
Ex 8 revisited) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store
What is the probability that at least one bulb is defective
P(at least one) = 1 - P(DC amp DC)
0975
Ex 9) For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something
P(at least one winning symbol) =
1 ndash P(no winning symbols) 1 - 96 = 4686
Rule 7 Conditional Probability
bull A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
Law of Large Numbers
bull As the number of repetitions of a chance experiment increase the difference between the relative frequency of occurrence for an event and the true probability approaches zero
Basic Rules of ProbabilityRule 1 Legitimate Values
For any event E 0 lt P(E) lt 1
Rule 2 Sample spaceIf S is the sample space P(S) = 1
Rule 3 Complement
For any event E P(E) + P(not E) = 1
Complement of an eventhellip
Ec
Independentbull Two events are independent if knowing that
one will occur (or has occurred) does not change the probability that the other occurs
So Chuck what are you going to do next
year after graduation
Oh I plan on going to a culinary school to become a chef ndash I
just love to cook
Teacher to student in school
Teacher to student in front of parent
Oh remember ndash Irsquom going to college to
become an engineer
Are Chuckrsquos responses independent
Rule 4 Multiplication
If two events A amp B are independent
General rule
P(B) P(A) B) ampP(A
A)|P(B P(A) B) ampP(A
)( BAP
Independent)( BAP
Yes
)()( BPAP
What does this mean
Ex 1) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that both bulbs are defective
Can you assume they are independent
00250505D) amp P(D
)( BAP
Independent)( BAP
Yes No
)()( BPAP )|()( ABPAP
Ex 2) There are seven girls and eight boys in a math class The teacher selects two students at random to answer questions on the board What is the probability that both students are girls
Are these events independent
2146
157
G) amp P(G
Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades
Are the cards independent NO
P(A amp B) = P(A) P(B|A)
P(Spade amp Spade) = 14 1251 = 117
The probability of getting a spade given that a spade has already been drawn
Rule 5 AdditionIf two events E amp F are disjoint
P(E or F) = P(E) + P(F)
(General) If two events E amp F are not disjoint
P(E or F) = P(E) + P(F) ndash P(E amp F)
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota (Note these are not simple events since there are many types of each brand) Suppose that P(H) = 25 P(N) = 18 P(T) = 14
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAILS Roll is LESS THAN 3
P (tails or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
Rule 6 At least one
The probability that at least one outcome happens is 1 minus the probability that no outcomes happen
P(at least 1) = 1 ndash P(none)
Ex 8 revisited) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store
What is the probability that at least one bulb is defective
P(at least one) = 1 - P(DC amp DC)
0975
Ex 9) For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something
P(at least one winning symbol) =
1 ndash P(no winning symbols) 1 - 96 = 4686
Rule 7 Conditional Probability
bull A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
Basic Rules of ProbabilityRule 1 Legitimate Values
For any event E 0 lt P(E) lt 1
Rule 2 Sample spaceIf S is the sample space P(S) = 1
Rule 3 Complement
For any event E P(E) + P(not E) = 1
Complement of an eventhellip
Ec
Independentbull Two events are independent if knowing that
one will occur (or has occurred) does not change the probability that the other occurs
So Chuck what are you going to do next
year after graduation
Oh I plan on going to a culinary school to become a chef ndash I
just love to cook
Teacher to student in school
Teacher to student in front of parent
Oh remember ndash Irsquom going to college to
become an engineer
Are Chuckrsquos responses independent
Rule 4 Multiplication
If two events A amp B are independent
General rule
P(B) P(A) B) ampP(A
A)|P(B P(A) B) ampP(A
)( BAP
Independent)( BAP
Yes
)()( BPAP
What does this mean
Ex 1) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that both bulbs are defective
Can you assume they are independent
00250505D) amp P(D
)( BAP
Independent)( BAP
Yes No
)()( BPAP )|()( ABPAP
Ex 2) There are seven girls and eight boys in a math class The teacher selects two students at random to answer questions on the board What is the probability that both students are girls
Are these events independent
2146
157
G) amp P(G
Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades
Are the cards independent NO
P(A amp B) = P(A) P(B|A)
P(Spade amp Spade) = 14 1251 = 117
The probability of getting a spade given that a spade has already been drawn
Rule 5 AdditionIf two events E amp F are disjoint
P(E or F) = P(E) + P(F)
(General) If two events E amp F are not disjoint
P(E or F) = P(E) + P(F) ndash P(E amp F)
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota (Note these are not simple events since there are many types of each brand) Suppose that P(H) = 25 P(N) = 18 P(T) = 14
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAILS Roll is LESS THAN 3
P (tails or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
Rule 6 At least one
The probability that at least one outcome happens is 1 minus the probability that no outcomes happen
P(at least 1) = 1 ndash P(none)
Ex 8 revisited) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store
What is the probability that at least one bulb is defective
P(at least one) = 1 - P(DC amp DC)
0975
Ex 9) For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something
P(at least one winning symbol) =
1 ndash P(no winning symbols) 1 - 96 = 4686
Rule 7 Conditional Probability
bull A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
Rule 3 Complement
For any event E P(E) + P(not E) = 1
Complement of an eventhellip
Ec
Independentbull Two events are independent if knowing that
one will occur (or has occurred) does not change the probability that the other occurs
So Chuck what are you going to do next
year after graduation
Oh I plan on going to a culinary school to become a chef ndash I
just love to cook
Teacher to student in school
Teacher to student in front of parent
Oh remember ndash Irsquom going to college to
become an engineer
Are Chuckrsquos responses independent
Rule 4 Multiplication
If two events A amp B are independent
General rule
P(B) P(A) B) ampP(A
A)|P(B P(A) B) ampP(A
)( BAP
Independent)( BAP
Yes
)()( BPAP
What does this mean
Ex 1) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that both bulbs are defective
Can you assume they are independent
00250505D) amp P(D
)( BAP
Independent)( BAP
Yes No
)()( BPAP )|()( ABPAP
Ex 2) There are seven girls and eight boys in a math class The teacher selects two students at random to answer questions on the board What is the probability that both students are girls
Are these events independent
2146
157
G) amp P(G
Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades
Are the cards independent NO
P(A amp B) = P(A) P(B|A)
P(Spade amp Spade) = 14 1251 = 117
The probability of getting a spade given that a spade has already been drawn
Rule 5 AdditionIf two events E amp F are disjoint
P(E or F) = P(E) + P(F)
(General) If two events E amp F are not disjoint
P(E or F) = P(E) + P(F) ndash P(E amp F)
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota (Note these are not simple events since there are many types of each brand) Suppose that P(H) = 25 P(N) = 18 P(T) = 14
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAILS Roll is LESS THAN 3
P (tails or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
Rule 6 At least one
The probability that at least one outcome happens is 1 minus the probability that no outcomes happen
P(at least 1) = 1 ndash P(none)
Ex 8 revisited) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store
What is the probability that at least one bulb is defective
P(at least one) = 1 - P(DC amp DC)
0975
Ex 9) For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something
P(at least one winning symbol) =
1 ndash P(no winning symbols) 1 - 96 = 4686
Rule 7 Conditional Probability
bull A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
Independentbull Two events are independent if knowing that
one will occur (or has occurred) does not change the probability that the other occurs
So Chuck what are you going to do next
year after graduation
Oh I plan on going to a culinary school to become a chef ndash I
just love to cook
Teacher to student in school
Teacher to student in front of parent
Oh remember ndash Irsquom going to college to
become an engineer
Are Chuckrsquos responses independent
Rule 4 Multiplication
If two events A amp B are independent
General rule
P(B) P(A) B) ampP(A
A)|P(B P(A) B) ampP(A
)( BAP
Independent)( BAP
Yes
)()( BPAP
What does this mean
Ex 1) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that both bulbs are defective
Can you assume they are independent
00250505D) amp P(D
)( BAP
Independent)( BAP
Yes No
)()( BPAP )|()( ABPAP
Ex 2) There are seven girls and eight boys in a math class The teacher selects two students at random to answer questions on the board What is the probability that both students are girls
Are these events independent
2146
157
G) amp P(G
Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades
Are the cards independent NO
P(A amp B) = P(A) P(B|A)
P(Spade amp Spade) = 14 1251 = 117
The probability of getting a spade given that a spade has already been drawn
Rule 5 AdditionIf two events E amp F are disjoint
P(E or F) = P(E) + P(F)
(General) If two events E amp F are not disjoint
P(E or F) = P(E) + P(F) ndash P(E amp F)
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota (Note these are not simple events since there are many types of each brand) Suppose that P(H) = 25 P(N) = 18 P(T) = 14
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAILS Roll is LESS THAN 3
P (tails or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
Rule 6 At least one
The probability that at least one outcome happens is 1 minus the probability that no outcomes happen
P(at least 1) = 1 ndash P(none)
Ex 8 revisited) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store
What is the probability that at least one bulb is defective
P(at least one) = 1 - P(DC amp DC)
0975
Ex 9) For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something
P(at least one winning symbol) =
1 ndash P(no winning symbols) 1 - 96 = 4686
Rule 7 Conditional Probability
bull A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
Rule 4 Multiplication
If two events A amp B are independent
General rule
P(B) P(A) B) ampP(A
A)|P(B P(A) B) ampP(A
)( BAP
Independent)( BAP
Yes
)()( BPAP
What does this mean
Ex 1) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that both bulbs are defective
Can you assume they are independent
00250505D) amp P(D
)( BAP
Independent)( BAP
Yes No
)()( BPAP )|()( ABPAP
Ex 2) There are seven girls and eight boys in a math class The teacher selects two students at random to answer questions on the board What is the probability that both students are girls
Are these events independent
2146
157
G) amp P(G
Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades
Are the cards independent NO
P(A amp B) = P(A) P(B|A)
P(Spade amp Spade) = 14 1251 = 117
The probability of getting a spade given that a spade has already been drawn
Rule 5 AdditionIf two events E amp F are disjoint
P(E or F) = P(E) + P(F)
(General) If two events E amp F are not disjoint
P(E or F) = P(E) + P(F) ndash P(E amp F)
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota (Note these are not simple events since there are many types of each brand) Suppose that P(H) = 25 P(N) = 18 P(T) = 14
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAILS Roll is LESS THAN 3
P (tails or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
Rule 6 At least one
The probability that at least one outcome happens is 1 minus the probability that no outcomes happen
P(at least 1) = 1 ndash P(none)
Ex 8 revisited) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store
What is the probability that at least one bulb is defective
P(at least one) = 1 - P(DC amp DC)
0975
Ex 9) For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something
P(at least one winning symbol) =
1 ndash P(no winning symbols) 1 - 96 = 4686
Rule 7 Conditional Probability
bull A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
)( BAP
Independent)( BAP
Yes
)()( BPAP
What does this mean
Ex 1) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that both bulbs are defective
Can you assume they are independent
00250505D) amp P(D
)( BAP
Independent)( BAP
Yes No
)()( BPAP )|()( ABPAP
Ex 2) There are seven girls and eight boys in a math class The teacher selects two students at random to answer questions on the board What is the probability that both students are girls
Are these events independent
2146
157
G) amp P(G
Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades
Are the cards independent NO
P(A amp B) = P(A) P(B|A)
P(Spade amp Spade) = 14 1251 = 117
The probability of getting a spade given that a spade has already been drawn
Rule 5 AdditionIf two events E amp F are disjoint
P(E or F) = P(E) + P(F)
(General) If two events E amp F are not disjoint
P(E or F) = P(E) + P(F) ndash P(E amp F)
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota (Note these are not simple events since there are many types of each brand) Suppose that P(H) = 25 P(N) = 18 P(T) = 14
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAILS Roll is LESS THAN 3
P (tails or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
Rule 6 At least one
The probability that at least one outcome happens is 1 minus the probability that no outcomes happen
P(at least 1) = 1 ndash P(none)
Ex 8 revisited) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store
What is the probability that at least one bulb is defective
P(at least one) = 1 - P(DC amp DC)
0975
Ex 9) For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something
P(at least one winning symbol) =
1 ndash P(no winning symbols) 1 - 96 = 4686
Rule 7 Conditional Probability
bull A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
Ex 1) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that both bulbs are defective
Can you assume they are independent
00250505D) amp P(D
)( BAP
Independent)( BAP
Yes No
)()( BPAP )|()( ABPAP
Ex 2) There are seven girls and eight boys in a math class The teacher selects two students at random to answer questions on the board What is the probability that both students are girls
Are these events independent
2146
157
G) amp P(G
Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades
Are the cards independent NO
P(A amp B) = P(A) P(B|A)
P(Spade amp Spade) = 14 1251 = 117
The probability of getting a spade given that a spade has already been drawn
Rule 5 AdditionIf two events E amp F are disjoint
P(E or F) = P(E) + P(F)
(General) If two events E amp F are not disjoint
P(E or F) = P(E) + P(F) ndash P(E amp F)
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota (Note these are not simple events since there are many types of each brand) Suppose that P(H) = 25 P(N) = 18 P(T) = 14
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAILS Roll is LESS THAN 3
P (tails or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
Rule 6 At least one
The probability that at least one outcome happens is 1 minus the probability that no outcomes happen
P(at least 1) = 1 ndash P(none)
Ex 8 revisited) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store
What is the probability that at least one bulb is defective
P(at least one) = 1 - P(DC amp DC)
0975
Ex 9) For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something
P(at least one winning symbol) =
1 ndash P(no winning symbols) 1 - 96 = 4686
Rule 7 Conditional Probability
bull A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
)( BAP
Independent)( BAP
Yes No
)()( BPAP )|()( ABPAP
Ex 2) There are seven girls and eight boys in a math class The teacher selects two students at random to answer questions on the board What is the probability that both students are girls
Are these events independent
2146
157
G) amp P(G
Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades
Are the cards independent NO
P(A amp B) = P(A) P(B|A)
P(Spade amp Spade) = 14 1251 = 117
The probability of getting a spade given that a spade has already been drawn
Rule 5 AdditionIf two events E amp F are disjoint
P(E or F) = P(E) + P(F)
(General) If two events E amp F are not disjoint
P(E or F) = P(E) + P(F) ndash P(E amp F)
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota (Note these are not simple events since there are many types of each brand) Suppose that P(H) = 25 P(N) = 18 P(T) = 14
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAILS Roll is LESS THAN 3
P (tails or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
Rule 6 At least one
The probability that at least one outcome happens is 1 minus the probability that no outcomes happen
P(at least 1) = 1 ndash P(none)
Ex 8 revisited) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store
What is the probability that at least one bulb is defective
P(at least one) = 1 - P(DC amp DC)
0975
Ex 9) For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something
P(at least one winning symbol) =
1 ndash P(no winning symbols) 1 - 96 = 4686
Rule 7 Conditional Probability
bull A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
Ex 2) There are seven girls and eight boys in a math class The teacher selects two students at random to answer questions on the board What is the probability that both students are girls
Are these events independent
2146
157
G) amp P(G
Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades
Are the cards independent NO
P(A amp B) = P(A) P(B|A)
P(Spade amp Spade) = 14 1251 = 117
The probability of getting a spade given that a spade has already been drawn
Rule 5 AdditionIf two events E amp F are disjoint
P(E or F) = P(E) + P(F)
(General) If two events E amp F are not disjoint
P(E or F) = P(E) + P(F) ndash P(E amp F)
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota (Note these are not simple events since there are many types of each brand) Suppose that P(H) = 25 P(N) = 18 P(T) = 14
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAILS Roll is LESS THAN 3
P (tails or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
Rule 6 At least one
The probability that at least one outcome happens is 1 minus the probability that no outcomes happen
P(at least 1) = 1 ndash P(none)
Ex 8 revisited) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store
What is the probability that at least one bulb is defective
P(at least one) = 1 - P(DC amp DC)
0975
Ex 9) For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something
P(at least one winning symbol) =
1 ndash P(no winning symbols) 1 - 96 = 4686
Rule 7 Conditional Probability
bull A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades
Are the cards independent NO
P(A amp B) = P(A) P(B|A)
P(Spade amp Spade) = 14 1251 = 117
The probability of getting a spade given that a spade has already been drawn
Rule 5 AdditionIf two events E amp F are disjoint
P(E or F) = P(E) + P(F)
(General) If two events E amp F are not disjoint
P(E or F) = P(E) + P(F) ndash P(E amp F)
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota (Note these are not simple events since there are many types of each brand) Suppose that P(H) = 25 P(N) = 18 P(T) = 14
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAILS Roll is LESS THAN 3
P (tails or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
Rule 6 At least one
The probability that at least one outcome happens is 1 minus the probability that no outcomes happen
P(at least 1) = 1 ndash P(none)
Ex 8 revisited) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store
What is the probability that at least one bulb is defective
P(at least one) = 1 - P(DC amp DC)
0975
Ex 9) For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something
P(at least one winning symbol) =
1 ndash P(no winning symbols) 1 - 96 = 4686
Rule 7 Conditional Probability
bull A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
Rule 5 AdditionIf two events E amp F are disjoint
P(E or F) = P(E) + P(F)
(General) If two events E amp F are not disjoint
P(E or F) = P(E) + P(F) ndash P(E amp F)
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota (Note these are not simple events since there are many types of each brand) Suppose that P(H) = 25 P(N) = 18 P(T) = 14
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAILS Roll is LESS THAN 3
P (tails or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
Rule 6 At least one
The probability that at least one outcome happens is 1 minus the probability that no outcomes happen
P(at least 1) = 1 ndash P(none)
Ex 8 revisited) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store
What is the probability that at least one bulb is defective
P(at least one) = 1 - P(DC amp DC)
0975
Ex 9) For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something
P(at least one winning symbol) =
1 ndash P(no winning symbols) 1 - 96 = 4686
Rule 7 Conditional Probability
bull A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota (Note these are not simple events since there are many types of each brand) Suppose that P(H) = 25 P(N) = 18 P(T) = 14
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAILS Roll is LESS THAN 3
P (tails or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
Rule 6 At least one
The probability that at least one outcome happens is 1 minus the probability that no outcomes happen
P(at least 1) = 1 ndash P(none)
Ex 8 revisited) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store
What is the probability that at least one bulb is defective
P(at least one) = 1 - P(DC amp DC)
0975
Ex 9) For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something
P(at least one winning symbol) =
1 ndash P(no winning symbols) 1 - 96 = 4686
Rule 7 Conditional Probability
bull A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota (Note these are not simple events since there are many types of each brand) Suppose that P(H) = 25 P(N) = 18 P(T) = 14
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAILS Roll is LESS THAN 3
P (tails or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
Rule 6 At least one
The probability that at least one outcome happens is 1 minus the probability that no outcomes happen
P(at least 1) = 1 ndash P(none)
Ex 8 revisited) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store
What is the probability that at least one bulb is defective
P(at least one) = 1 - P(DC amp DC)
0975
Ex 9) For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something
P(at least one winning symbol) =
1 ndash P(no winning symbols) 1 - 96 = 4686
Rule 7 Conditional Probability
bull A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAILS Roll is LESS THAN 3
P (tails or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
Rule 6 At least one
The probability that at least one outcome happens is 1 minus the probability that no outcomes happen
P(at least 1) = 1 ndash P(none)
Ex 8 revisited) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store
What is the probability that at least one bulb is defective
P(at least one) = 1 - P(DC amp DC)
0975
Ex 9) For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something
P(at least one winning symbol) =
1 ndash P(no winning symbols) 1 - 96 = 4686
Rule 7 Conditional Probability
bull A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAILS Roll is LESS THAN 3
P (tails or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
Rule 6 At least one
The probability that at least one outcome happens is 1 minus the probability that no outcomes happen
P(at least 1) = 1 ndash P(none)
Ex 8 revisited) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store
What is the probability that at least one bulb is defective
P(at least one) = 1 - P(DC amp DC)
0975
Ex 9) For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something
P(at least one winning symbol) =
1 ndash P(no winning symbols) 1 - 96 = 4686
Rule 7 Conditional Probability
bull A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAILS Roll is LESS THAN 3
P (tails or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
Rule 6 At least one
The probability that at least one outcome happens is 1 minus the probability that no outcomes happen
P(at least 1) = 1 ndash P(none)
Ex 8 revisited) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store
What is the probability that at least one bulb is defective
P(at least one) = 1 - P(DC amp DC)
0975
Ex 9) For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something
P(at least one winning symbol) =
1 ndash P(no winning symbols) 1 - 96 = 4686
Rule 7 Conditional Probability
bull A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAILS Roll is LESS THAN 3
P (tails or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
Rule 6 At least one
The probability that at least one outcome happens is 1 minus the probability that no outcomes happen
P(at least 1) = 1 ndash P(none)
Ex 8 revisited) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store
What is the probability that at least one bulb is defective
P(at least one) = 1 - P(DC amp DC)
0975
Ex 9) For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something
P(at least one winning symbol) =
1 ndash P(no winning symbols) 1 - 96 = 4686
Rule 7 Conditional Probability
bull A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAILS Roll is LESS THAN 3
P (tails or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
Rule 6 At least one
The probability that at least one outcome happens is 1 minus the probability that no outcomes happen
P(at least 1) = 1 ndash P(none)
Ex 8 revisited) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store
What is the probability that at least one bulb is defective
P(at least one) = 1 - P(DC amp DC)
0975
Ex 9) For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something
P(at least one winning symbol) =
1 ndash P(no winning symbols) 1 - 96 = 4686
Rule 7 Conditional Probability
bull A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
Rule 6 At least one
The probability that at least one outcome happens is 1 minus the probability that no outcomes happen
P(at least 1) = 1 ndash P(none)
Ex 8 revisited) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store
What is the probability that at least one bulb is defective
P(at least one) = 1 - P(DC amp DC)
0975
Ex 9) For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something
P(at least one winning symbol) =
1 ndash P(no winning symbols) 1 - 96 = 4686
Rule 7 Conditional Probability
bull A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
Rule 6 At least one
The probability that at least one outcome happens is 1 minus the probability that no outcomes happen
P(at least 1) = 1 ndash P(none)
Ex 8 revisited) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store
What is the probability that at least one bulb is defective
P(at least one) = 1 - P(DC amp DC)
0975
Ex 9) For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something
P(at least one winning symbol) =
1 ndash P(no winning symbols) 1 - 96 = 4686
Rule 7 Conditional Probability
bull A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
Rule 6 At least one
The probability that at least one outcome happens is 1 minus the probability that no outcomes happen
P(at least 1) = 1 ndash P(none)
Ex 8 revisited) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store
What is the probability that at least one bulb is defective
P(at least one) = 1 - P(DC amp DC)
0975
Ex 9) For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something
P(at least one winning symbol) =
1 ndash P(no winning symbols) 1 - 96 = 4686
Rule 7 Conditional Probability
bull A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
Ex 8 revisited) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store
What is the probability that at least one bulb is defective
P(at least one) = 1 - P(DC amp DC)
0975
Ex 9) For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something
P(at least one winning symbol) =
1 ndash P(no winning symbols) 1 - 96 = 4686
Rule 7 Conditional Probability
bull A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
Ex 9) For a sales promotion the manufacturer places winning symbols under the caps of 10 of all Dr Pepper bottles You buy a six-pack What is the probability that you win something
P(at least one winning symbol) =
1 ndash P(no winning symbols) 1 - 96 = 4686
Rule 7 Conditional Probability
bull A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
Rule 7 Conditional Probability
bull A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
