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1
Chapter 3Probability
3-1 Fundamentals
3-2 Addition Rule
3-3 Multiplication Rule: Basics
3-4 Multiplication Rule: Complements and Conditional Probability
3-5 Counting Techniques
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Objectives develop sound understanding of
probability values used in subsequent chapters
develop basic skills necessary to solve simple probability problems
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General CommentsThis chapter tends to be the most difficult one encountered in the course
Homework note:Show setup of problem even if using calculator
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Experiment – Action being performed (book uses the word procedure)
Event (E) – A particular observation within the experiment
Sample space (S) - all possible events within the experiment
3-1 Fundamentals Definitions
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Notation
P - denotes a probability
A, B, ... - denote specific events
P (A) - denotes the probability of event A occurring
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Basic Rules for Computing Probability
Let A equal an Event
P(A) =number of outcomes favorable to
event “A”total possible experimental outcomes(sample space)
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Probability Limits
The probability of an impossible event is 0.
The probability of an event that is certain to occur is 1.
0 P(A) 1
Impossibleto occur
Certainto occur
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Possible Values for ProbabilitiesCertain
Likely
50-50 Chance
Unlikely
Impossible
1
0.5
0
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Unlikely Probabilities
Examples:• Winning the lottery• Being struck by lightning• 0.0000035892• 1 / 727,235
Typically any probability less than 0.05 is considered unlikely.
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Example: Roll a die and observe a 4? Find the probability.
What is the experiment? Roll a die
What is the event A? Observe a 4
What is the sample space? 1,2,3,4,5,6
Number of outcomes favorable to A is 1.
Number of total outcomes is 6.
What is P(A)? P(A) = 1 / 6 = 0.167
Similar to #4 on hw
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Example: Toss a coin 3 times and observe exactly 2 heads?
Experiment:
toss a coin 3 times
Event (A):
observe exactly 2 heads
Sample Space:
HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
Note there are 3 outcomes favorable to the event and 8 total outcomes
P(A) = 3 / 8 = 0.375
Test problem
Similar to #6 on HW
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Law of Large Numbers
As a procedure is repeated again and again, the probability of an event tends to approach the actual probability.
This is the reason to quit while your ahead when gambling!
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P(A)
Complementary Events
The complement of event A, denoted by A, consists of all outcomes in which event A does not occur.
P(A)(read “not A”)
or P(Ac)
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P(A) = 1 – P(Ac)
Complementary Events
Property of complementary events
Example: If the probability of something occurring is 1/6 what is the probability that it won’t occur?
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Example: A study of randomly selected American Airlines flights showed that 344 arrived on time and 56 arrived late, What is the probability of a flight arriving late?
Let A = late flight Ac = on time flight
P(Ac) = 344 /(344 + 56) = 344/400 = .86
P(A) = 1 – P(Ac) = 1 - .86 = 0.14 Similar to number 11 & 12 on hw
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Rounding Off Probabilities
give the exact fraction or decimal
orround off the final result to three significant digits
Examples:
•1/3 is exact and could be left as a fraction or rounded to .333
•0.00038795 would be rounded to 0.000388
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Compound Event – Any event combining 2 or more events
Notation – P(A or B) = P (event A occurs or event B occurs or they both occur)
General Rule – add the total ways A can occur and the total way B can occur but don’t double count
3-2 Addition Rule Definitions
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• Formal Addition Rule• P(A or B) = P(A) + P(B) - P(A and B)
where P(A and B) denotes the probability that A and B both occur at the same time.
• Alternate form
P(A B) = P(A) + P(B) – P(A B)
Compound Event
U
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Definition
Events A and B are mutually exclusive if they cannot occur
simultaneously.
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Definition
Total Area = 1 Total Area = 1
P(A) P(B) P(A) P(B)
P(A and B)
Non-overlapping EventsOverlapping Events
Not Mutually Exclusive
P(A or B) = P(A) + P(B) – P(A and B)
Mutually Exclusive
P(A or B) = P(A) + P(B)
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Applying the Addition Rule
P(A or B)
Addition Rule
AreA and Bmutuallyexclusive
?
P(A or B) = P(A)+ P(B) - P(A and B)
P(A or B) = P(A) + P(B)Yes
No
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Mutually Exclusive
Example: P(A) = 2/7 and P(B) = 3/7 , P(A or B) = 5/7, are A and B mutually exclusive? Why?
Test question
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Example: You have an URN with 2 green marbles, 3 red marbles and 4 white marbles
Let A = choose red marble and B = choose white marble
1. What is the probability of choosing a red marble? P(A) = 3/9
2. What is the probability of choosing a white marble? P(B) = 4/9
3. What is the probability of choosing a red or a white? P(B or A) = 3/9 + 4/9 = 7/9
Test Questions
Why are event A and B mutually exclusive events?
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Example:
A card is drawn from a deck of cards.
1. What is the probability that the card is an ace or jack?
P(ace) + P(jack) = 4/52 + 4/52 = 8/52
2. What is the probability that the card is an ace or heart?
P(ace) + P(heart) – P(ace of hearts) = 4/52 + 13/52 – 1/52 = 16/52
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Example: Toss a coin 3 times and observe all possibilities of the number of heads
Experiment: toss a coin 3 times
Events (A): observe exactly 0 heads
(B): observe exactly 1 head
(C): observe exactly 2 heads
(D): observe exactly 3 heads
Sample Space:
HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
Find the P(A) + P(B) + P(C) + P(D)
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Example: Toss a coin 3 times and observe all possibilities of the number of heads
Event P(event)
A = 0 heads 1/8
B = 1 head 3/8
C = 2 heads 3/8
D = 3 heads 1/8
Total 1
Probability distribution: Table of all possible events along with the probability of each event. The sum of all probabilities must sum to ONE.
Note: Events are mutually exclusive
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Example: Roll 2 dice and observe the sum
Experiment: roll 2 dice
Event (F): observe sum of 5
Sample Space: 36 elements
One 2 Two 3’s Three 4’s Four 5’s Five 6’s, One 12 Two 11’s Three 10’s Four 9’s Five 8’s
Six 7’s
Find the P(F)
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Example: Roll 2 dice and observe the sum
Event P(event)
A: Sum = 2 1/36
B: Sum = 3 2/36
And so on…….
Construct a probability distribution
Let’s Try #8 From the HW
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• Let A = select a man • Let B = select a girl
• P(A or B) = + = = 0.781
Men Women Boys Girls Totals
Survived 332 318 29 27 706
Died 1360 104 35 18 1517
Total 1692 422 64 45 2223
Contingency Table (Titanic Mortality)
* Mutually Exclusive *
16922223
452223
17372223
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• Let A = select a woman• Let B = select someone who died.
P(A or B) = (422 + 1517 - 104) / 2223 = 1835 / 2223 = 0.825
Men Women Boys Girls Totals
Survived 332 318 29 27 706
Died 1360 104 35 18 1517
Total 1692 422 64 45 2223
Contingency Table(Titanic Mortality)
* NOT Mutually Exclusive * Very similar to test problem
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• How could you define a probability distribution for this data?
Men Women Boys Girls Totals
Survived 332 318 29 27 706
Died 1360 104 35 18 1517
Total 1692 422 64 45 2223
Contingency Table (Titanic Mortality)
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Complementary EventsP(A) & P(Ac)
P(A) and P(Ac) are mutually exclusive
P(A) + P(Ac) = 1 (this has to be true)
P(A) = 1 - P(Ac)
P(Ac) = 1 – P(A)
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Venn Diagram for the Complement of Event A
Total Area = 1
P (A)
P (A) = 1 - P (A)
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Notation: P(A and B) = P(event A occurs in a first trial and event B occurs in a second trial)
Formal Rule
P(A and B) = P(A) • P(B) if independent (with replacement)
P(A and B) = P(A) • P(B A) if dependent (without replacement)
3-3 Multiplication Rule Definitions
will define later
35
Definitions Independent Events
Two events A and B are independent if the occurrence of one does not affect the probability of the occurrence of the other.
Dependent EventsIf A and B are not independent, they are said to be dependent.
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TaTbTcTdTeFaFbFcFdFe
a
b
c
d
e
a
b
c
d
e
T
F
P(T) = P(c) = P(T and c) =
Tree Diagram of Test Answers
12
15
110
37
P (both correct) = P (T and c) 1
10 1 2
1 5
= •
MultiplicationRule
INDEPENDENT EVENTS
38
Choose 2 marbles from an URN with 3 red marbles and 3 white marbles
Dependent – choose the 1st marble then choose the 2nd marble
Independent – choose the 1st marble, replace it, then choose the 2nd marble
Independence vs. Dependence
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P(B A) represents the probability of event B occurring after it is assumed that event A has already occurred (read B A as “B given A”).
= given
Notation for Conditional Probability
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Example: You have an URN with 3 red marbles and 4 white marbles
Let A = choose red marble and B = choose white marble
1. What is the probability of choosing a red marble? P(A)
2. What is the probability of choosing a white marble? P(B)
3. If two are chosen find the probability of choosing a white on the a second trial given a red marble was chosen 1st. P(B A)
a) Assume the 1st marble is replaced {independent}
b) Assume the 1st marble is not replaced {dependent}
Test Questions
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Example: You have an URN with 3 red marbles and 4 white marbles
Let’s change things a bit….
If two are chosen and we want to find the probability of choosing a white then choosing a red marble.
So if we let: A = choose red 1st and B = choose white 2nd then we need to find P(A and B)P(A and B)
The problem here is that calculating this probability depends on what happens on the first draw. We need a rule that helps us with this.
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Formal Multiplication Rule
P(A and B) = P(A) • P(B A)
If A and B are independent events, P(B A) is really the same as P(B). Will see this in the next section.
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Applying the Multiplication Rule
P(A and B)
Multiplication Rule
AreA and B
independent?
P(A and B) = P(A) • P(B A)
P(A and B) = P(A) • P(B)Yes
No
44
Example: You have an URN with 3 red marbles and 4 white marbles
Let A = choose red marble and B = choose white marble
If two are chosen find the probability of choosing a red then choosing a white marble. In other words find P(A and B)
a) Assume the 1st marble is replaced {independent}P(A and B) = P(A) • P(B)
b) Assume the 1st marble is not replaced {dependent}P(A and B) = P(A) • P(B A)
Use as an example for #6
Test Questions
45
Class Assignment – Part IYou have an URN with 3 red marbles, 7 white marbles,
and 1 green marbleLet A = choose red
B = choose whiteC = choose green
Find the following:
1. P(Ac), that is find P(not red)2. P(A or B)3. If two marbles chosen what is the probability that you
choose a white marble 2nd when a red marble was chosen first. This is, find P(B A)
4. If two marbles are chosen, find P(A and C) with replacement5. If two marbles are chosen, find P(A and C) without
replacement
Very Similar to test question #1
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Class Assignment – Part II
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Mutually Exclusive vs. Independent Events
Mutually Exclusive EventsP(A or B) = P(A) + P(B)
Independent Events P(A and B) = P(A) • P(B)
Example: if P(A) = .3, P(B)=.4, P(A or B)=.7, and P(A and B) = .12, what can you say about A and B?
Note: Test Question
48
• Select two– Find P(2 women) = 422/2223 x 421/2222– Find P(2 that died) = 1517/2223 x 1516/2222
• Select one– Find P(woman and died) = 104/2223– Find P(Boy and survived) = 29 / 2223
Men Women Boys Girls Totals
Survived 332 318 29 27 706
Died 1360 104 35 18 1517
Total 1692 422 64 45 2223
Contingency Table(Titanic Mortality)
49
3-4 Topics
Probability of “at least one”
More on conditional probability
Test for independence
50
Probability of ‘At Least One’
‘At least one’ is equivalent to ‘one or more’.
The complement of getting at least one item of a particular type is that you get
no items of that type.
If P(A) = P(getting at least one), then
P(A) = 1 - P(Ac)
where P(Ac) = P(getting none)
51
Probability of ‘At Least One’
Find the probability of a getting at least 1 head if you toss a coin 4 times.
P(A) = 1 - P(Ac)
where P(A) is P(no heads)
P(Ac) = (0.5)(0.5)(0.5 )(0.5) = 0.0625
P(A) = 1 - 0.0625 = 0.9375
52
Conditional Probability
P(A and B) = P(A) • P(B|A)Divide both sides by P(A)
Formal definition for conditional probability
P(B|A) =P(A and B)
P(A)
53
If P(B|A) = P(B)
then the occurrence of A has no effect on the probability of event B; that is, A and B are independent events.
or
If P(A and B) = P(A) • P(B)
then A and B are independent events. (with replacement)
Testing for Independence
Example: if A and B are independent, find P(A and B) if P(A) = 0.3 and P(B) = 0.6 (test question)
54
3-5 Countingfundamental counting rule
two events (“mn” rule)
multiple events (nr rule)
permutationsfactorial rule
different items
not all items different
combinations
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Fundamental Counting Rule(‘mn” rule)
If one event can occur m ways and the second event can occur n ways, the events together can occur a total of m • n ways.
Example 1:Example 1: How many ways can your order a meal with 3 main course choices and 4 deserts? First list then use rule.
Main Courses
Tacos (T)
Pasta (P)
Liver & Onions (LO)
Deserts
Ice Cream (IC)
Jello (J)
Cake ©
Fruit (F)
56
Fundamental Counting Rule(nr rule)
If one event that can occur n ways is repeated r times, the events together
can occur a total of nr ways.
Example 2: Example 2: How many outcomes are possible when tossing a coin 3 times? First list then use rule.
Example 3: Example 3: How many outcomes are possible when tossing a coin 20 times? Would you care to list all the outcomes this time?
57
The factorial symbol ! denotes the product of decreasing positive whole numbers.
n! = n (n-1) (n-2) (n-3) • • • • • (3) (2) (1)
Special Definition: 0! = 1
Find the ! key on your calculator
Notation
58
A collection of n different items can be arranged in order n! different ways.
Factorial Rule
Example:Example: How many ways can you order the letters A, B, C? List first then use rule.
Note: actually a special type of permutation, will define next
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Compare the NR and Factorial Rule
Example:Example: How many ways can you order the letters A, B, C?
a) NR _______ _________ ________ (with replacement)
b) N! _______ _________ ________ (without replacement)
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n is the number of available items (without replacement)
r is the number of items to be selected
the number of permutations (or sequences) is
Permutations Rule(when items are all different)
Order matters
Pn r = (n - r)!n!
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Example: Eight men enter a race. In how many ways can the first 4 positions be determined?
Permutations Rule(when items are all different)
62
Permutations Rule( when some items are identical )
If there are n items with n1 alike, n2 alike, . .
. . nk alike, the number of permutations is
n1! . n2! .. . . . . . . nk! n!
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Permutations Rule( when some items are identical )
1. How many ways can you arrange the word statistics? Or Mississippi?
Examples:
2. How many ways can you arrange 3 green marbles and 4 red marbles?
Test Question
64
Permutations Rule
Factorial rule is special case
n! = n nP
Can you show this is true?
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n different items
r items to be selected
different orders of the same items are not counted (order doesn’t matter)
the number of combinations is
(n - r )! r!n!
nCr =
Combinations Rule
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TI-83 CalculatorCalculate n! , nPr, nCr
1. Enter the value for n
2. Press Math
3. Cursor over to Prb
4. Choose 2: nPr or 3: nCr or 4: n! as required
5a. Press Enter for the n! case
5b. Enter the value for “r” for the nPr and nCr
cases
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Pick five numbers from 1 to 47 and a MEGA number from 1 to 27
Pick five numbers from 1 to 56 and a MEGA number from 1 to 46
Note: game has 2 separate sets of numbers
68
Example: Find the probability of winning the Pennsylvania Super 6 lotto. Select 6 numbers from 69.
What’s the probably of getting 5 of 6? 4 of 6?, etc. (see lottery handout)
What’s the probability if you have to get all 6 numbers in a specified order?
Combinations Rule
69
Recall previous example?
Experiment: Toss a coin 3 times
Event (A): Observe 2 heads
Sample Space:
HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
Note there are 3 outcomes favorable to the event and 8 total outcomes
P(A) = 3 / 8 = 0.375
Test problem
70
Let’s take a different approach
Experiment: Toss a coin 3 times
Event (A): Observe 2 heads
There are 23 possible outcomes
and 3C2 ways to get 2 heads
P(A) = 3C2 / 23 = 3 / 8 = 0.375
Test problem
71
Example:
Experiment: Toss a coin 6 times
Event (A): Observe 4 heads
There are 26 possible outcomes
and 6C4 ways to get 4 heads
P(A) = 6C4 / 26 = 15 / 64 = 0.234
Test problem
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