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5- 2Chapter FiveA Survey of Probability A Survey of Probability
ConceptsConceptsGOALSWhen you have completed this chapter, you will be able to:
ONEDefine probability.
TWO Describe the classical, empirical, and subjective approaches to probability.
THREEUnderstand the terms: experiment, event, outcome, permutations, and combinations.
Goals
5- 3Chapter Five continued
A Survey of Probability A Survey of Probability ConceptsConceptsGOALS
When you have completed this chapter, you will be able to:FOURDefine the terms: conditional probability and joint probability.
FIVE Calculate probabilities using the rules of addition and the rules of multiplication.
SIXUse a tree diagram to organize and compute probabilities.
Goals
5- 6
Definitions continued
An EventEvent is the collection of one or more outcomes of an experiment.
An OutcomeOutcome is the particular result of an experiment.
Experiment:Experiment: A fair die is cast. A fair die is cast.
Possible outcomes: The Possible outcomes: The numbers 1, 2, 3, 4, 5, 6 numbers 1, 2, 3, 4, 5, 6
One possible event: The One possible event: The occurrence of an even occurrence of an even number. That is, we collect number. That is, we collect the outcomes 2, 4, and 6.the outcomes 2, 4, and 6.
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Definitions continued
There are three definitions of probability: classical, empirical, and subjective.
The
ClassicalClassical definition
applies when there are n
equally likely outcomes.
The EmpiricalEmpirical definition applies when the number of times the event happens is divided by the number of observations.
SubjectiveSubjective probability is based on whatever information is available.
5- 8
Mutually Exclusive Events
Events are Mutually Mutually ExclusiveExclusive if the occurrence of any one event means that none of the others can occur at the same time.
Mutually exclusive: Mutually exclusive: Rolling a 2 precludes Rolling a 2 precludes rolling a 1, 3, 4, 5, 6 rolling a 1, 3, 4, 5, 6 on the same roll.on the same roll.
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Collectively Exhaustive Events
Events are Collectively ExhaustiveCollectively Exhaustive if at least one of the events must occur when an experiment is conducted.
5- 10
Example 2
155.01200
186)( AP
Throughout her teaching career Professor Jones has awarded 186 A’s out of 1,200 students. What is the probability that a student in her section this semester will receive an A?
This is an example of the empirical definition of probability.
To find the probability a selected student earned an A:
5- 11
Subjective Probability
Examples of subjective probability are:
estimating the probability the Washington Redskins will win the Super Bowl this year.
estimating the probability mortgage rates for home loans will top 8 percent.
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Basic Rules of ProbabilityP(A or B) = P(A) + P(B)
If two events A and B are mutually
exclusive, the
Special Rule of Special Rule of AdditionAddition states that the
probability of A or B occurring equals the sum of
their respective probabilities.
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Example 3
Arrival Frequency
Early 100
On Time 800
Late 75
Canceled 25
Total 1000
New England Commuter Airways recently supplied the following information on their commuter flights from Boston to New York:
5- 14
Example 3 continued
The probability that a flight is either early or late is:
P(A or B) = P(A) + P(B) = .10 + .075 =.175.
If A is the event that a flight arrives early, then P(A) = 100/1000 = .10.
If B is the event that a flight arrives late, then P(B) = 75/1000 = .075.
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The Complement Rule
If P(A) is the probability of event A and P(~A) is the complement of A,
P(A) + P(~A) = 1 or P(A) = 1 - P(~A).
The Complement RuleComplement Rule is used to determine the probability of an event occurring by subtracting the probability of the event not occurring from 1.
5- 16
The Complement Rule continued
A~~AA
A Venn DiagramVenn Diagram illustrating the complement rule would appear as:
5- 17
Example 4
If D is the event that a flight is canceled, then
P(D) = 25/1000 = .025.
Recall example 3. Use the complement rule to find the probability of an early (A) or a late (B) flight
If C is the event that a flight arrives on time, then
P(C) = 800/1000 = .8.
5- 18
Example 4 continued
CC.8.8
DD.025.025
~(C or D) = (A or B)~(C or D) = (A or B) .175.175
P(A or B) = 1 - P(C or D) = 1 - [.8 +.025] =.175
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The General Rule of Addition
If A and B are two events that are not mutually exclusive, then P(A or B) is given by the following formula:
P(A or B) = P(A) + P(B) - P(A and B)
The General Rule of Addition
5- 21
EXAMPLE 5
StereoStereo 320320
BothBoth 100100
TVTV175175
In a sample of 500 students, 320 said they had a stereo, 175 said they had a TV, and 100 said they had both. 5 said they had neither.
NeitherNeither 55
5- 22
Example 5 continued
P(S or TV) = P(S) + P(TV) - P(S and TV)
= 320/500 + 175/500 – 100/500
= .79.
P(S and TV) = 100/500
= .20
If a student is selected at random, what is the probability that the student has only a stereo or TV? What is the probability that the student has both a stereo and TV?
5- 23
Joint Probability
A Joint ProbabilityJoint Probability measures the likelihood that two or more events will happen concurrently.
An example would be the event that a student has both a stereo and TV in his or her dorm room.
5- 24
Special Rule of Multiplication
Two events A and B are independent if the occurrence of one has no effect on the probability of the occurrence of the other.
This rule is written: P(A and B) = P(A)P(B)
The Special Rule of MultiplicationSpecial Rule of Multiplication requires that two events A and B are
independent.
5- 25
Example 6
5-year stock prices
0
5
10
15
20
25
30
35
40
45
1 2 3 4 5
Year
Stoc
k pr
ice
$
IBM
GE
P(IBM and GE) = (.5)(.7) = .35
Chris owns two stocks, IBM and General Electric (GE). The probability that IBM stock will increase in value next year is .5 and the probability that GE stock will increase in value next year is .7. Assume the two stocks are independent. What is the probability that both stocks will increase in value next year?
5- 26
Example 6 continued
P(at least one)
= P(IBM but not GE)
+ P(GE but not IBM)
+ P(IBM and GE)
W h at is th e
p ro b a b ility th a t a t lea st
o n e o f th ese sto ck s in creases in v a lu e in
th e n ex t y ear? T h is m ea n s th a t
e ith er o n e ca n in crease or
b o th .
(.5)(1-.7) + (.7)(1-.5) + (.7)(.5) = .85
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Conditional Probability
The probability of event A occurring given that the event B has occurred is written P(A|B).
A Conditional ProbabilityConditional Probability is the probability of a particular event occurring, given that another event has occurred.
5- 28
General Multiplication Rule
It states that for two events A and B, the joint probability that both events will happen is found by multiplying the probability that event A will happen by the conditional probability of B given that A has occurred.
The General General Rule of Rule of
MultiplicationMultiplication is used to find the joint probability that two events will occur.
5- 29
General Multiplication Rule
The joint probability, P(A and B), is given by the
following formula:
P(A and B) = P(A)P(B/A)
or P(A and B) =
P(B)P(A/B)
5- 30
Example 7
Major Male Female Total
Accounting 170 110 280
Finance 120 100 220
Marketing 160 70 230
Management 150 120 270
Total 600 400 1000
The Dean of the School of Business at Owens University collected the following information about undergraduate students in her college:
5- 31
Example 7 continued
P(A|F) = P(A and F)/P(F)
= [110/1000]/[400/1000] = .275
If a student is selected at random, what is the probability that the student is a female (F) accounting major (A)?
P(A and F) = 110/1000.
Given that the student is a female, what is the probability that she is an accounting major?
5- 32
Tree Diagrams
Example 8: In a bag containing 7 red chips and 5 blue chips you select 2 chips one after the other without replacement. Construct a tree diagram showing this information.
A Tree DiagramTree Diagram is useful for portraying conditional and joint probabilities. It is particularly useful for analyzing business decisions involving several stages.
5- 34
Some Principles of Counting
Example 10: Dr. Delong has 10 shirts and 8 ties. How many shirt and tie outfits does he have?
The Multiplication Multiplication FormulaFormula indicates that if there are m ways of doing one thing and n ways of doing another thing, there are m x n ways of doing both.
(10)(8) = 80
5- 35
Some Principles of Counting
)!(
!
rn
nP rn
A PermutationPermutation is any arrangement of r objects selected from n possible objects.
Note: The order of arrangement is important in permutations.
5- 36
Some Principles of Counting
)!(!
!
rnr
nCrn
A CombinationCombination is the number of ways to choose r objects from a group of n objects without regard to order.
5- 37
Example 11
792)!512(!5
!12512
C
There are 12 players on the Carolina Forest High School basketball team. Coach Thompson must pick five players among the twelve on the team to comprise the starting lineup. How many different groups are possible? (Order does not matter.)
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