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Rule

s

PROBABILITY – CONCEPTS OVERVIEW

A value between 0 and 1, inclusive, describing the relative possibility [chance or likelihood) an event to occur

Definitio

n

Objective

SubjectiveApproaches

1. Mutually exclusive [only one event can occur]P(A or B ) = P(A)+P(B)

2.Inclusive [both may occur]P(A or B) = P(A) +P(B) –P(A and B)

1. Special RuleP(A and B) = P(A)P(B)

2. General RuleP(A and B) = P(A)P(B/C)P(A and B and C) = P(A)P(P/A)P(C/A and B)

Multiplication [x]

Addition [ + ]

AA

AA

AA

AA

Posterior Probability[A revised probability based on additional information]

Bayes’ Theorem

Experiment, event, outcome, permutations, and combinations

Terms

PROBABILITY

5- 2Chapter FiveA Survey of Probability Concepts

GOALSWhen 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 continuedA Survey of Probability ConceptsGOALS

When you have completed this chapter, you will be able to:FOURDefine the terms: conditional probability and joint probability. FIVE Calculate probabilities applying the rules of addition and the rules of multiplication.SIXUse a tree diagram to organize and compute probabilities.SEVEN Calculate a probability using Bayes’ theorem. Goals

5- 4

Rule

s

PROBABILITY – CONCEPTS OVERVIEW

A value between 0 and 1, inclusive, describing the relative possibility [chance or likelihood) an event to occur

Definitio

n

Objective

SubjectiveApproaches

1. Mutually exclusive [only one event can occur]P(A or B ) = P(A)+P(B)

2.Inclusive [both may occur]P(A or B) = P(A) +P(B) –P(A and B)

1. Special RuleP(A and B) = P(A)P(B)

2. General RuleP(A and B) = P(A)P(B/C)P(A and B and C) = P(A)P(P/A)P(C/A and B)

Multiplication [x]

Addition [ + ]

AA

AA

AA

AA

Posterior Probability[A revised probability based on additional information]

Bayes’ Theorem

Experiment, event, outcome, permutations, and combinations

Terms

PROBABILITY

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DEFINITION OF PROBABILITYA value between 0 and 1, inclusive, describing the relative possibility [chance or likelihood) an event to occur

EXAMPLE:

Toasting of a coin A = Head , B = Tail

P(A) = Probability of getting “Head” = ½ = 0.5

P(B) = Probability of getting “Tail” = ½ - 0.5 P(A or B ) = ………………? = …….?

Note : Mutually Exclusive events = Head occurs Tail cannot occurs

5- 6

DEFINITION OF PROBABILITYA value between 0 and 1, inclusive, describing the relative possibility [chance or likelihood) an event to occur

EXAMPLE:

Company A and Company is Competing for a tender to be awarded by KUIS.

P(A) = Probability of Company A getting the tender = ……… ?

P(B) = Probability of Company B getting the tender = ……… ?

P(A or B ) = Probability of Company A or Company getting the tender = ………………? = …….?

Note : Mutually Exclusive events = Head occurs Tail cannot occurs

5- 7Definitions continued

There are three definitions of probability: classical, empirical, and subjective.

The Classical definition

applies when there are n

equally likely outcomes.

The Empirical definition applies when the number of times the event happens is divided by the number of observations.

Subjective probability is based on whatever information is available.

5- 8

Movie

5- 9

Definitions continued

An Event is the collection of one or more outcomes of an experiment.

An Outcome is the particular result of an experiment.

Experiment: A fair die is cast.

Possible outcomes: The numbers 1, 2, 3, 4, 5, 6

One possible event: The occurrence of an even number. That is, we collect the outcomes 2, 4, and 6.

5- 10

Mutually Exclusive Events

Events are Independent if the occurrence of one event does not affect the occurrence of another.

Events are Mutually Exclusive if the occurrence of any one event means that none of the others can occur at the same time.

Mutually exclusive: Rolling a 2 precludes rolling a 1, 3, 4, 5, 6 on the same roll.

Independence: Rolling a 2 on the first throw does not influence the probability of a 3 on the next throw. It is still a one in 6 chance.

5- 11

Collectively Exhaustive Events

Events are Collectively Exhaustive if at least one of the events must occur when an experiment is conducted.

5- 12

Example 2

155.01200186)( AP

Throughout his teaching career Mr. Shayaa Othman 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- 13

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, theSpecial Rule of

Addition states that the Probability of A or B

occurring equals the sum of their respective probabilities.

5- 15

Example 3

Arrival Frequency

Early 100

On Time 800

Late 75

Canceled 25

Total 1000

MALAYSIAN AIRLINE [MAS] recently supplied the following information on their commuter flights from Singapore to Kuala Lumpur:

5- 16

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.

5- 17

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 Rule is used to determine the probability of an event occurring by subtracting the probability of the event not occurring from 1.

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The Complement Rule continued

A ~A

A Venn Diagram illustrating the complement rule would appear as:

5- 19

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- 20

Example 4 continued

C.8

D.025

~(C or D) = (A or B) .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)

5- 22

The General Rule of Addition

A and BA

B

The Venn Diagram illustrates this rule:

5- 23

EXAMPLE 5

Stereo 320

Both 100

TV175

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.

5- 24

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- 25

Joint Probability

A Joint 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- 26

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 Multiplication requires that two events A and B are

independent.

5- 27

Example 6

5-year stock prices

05

1015202530354045

1 2 3 4 5Year

Stoc

k pr

ice

$

IBMGE

P(PROTON and TM) = (.5)(.7) = .35

Ahmad owns two stocks, PROTON and Telekom Malaysia (TM). The probability that PROTON stock will increase in value next year is .5 and the probability that TM 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?

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P(at least one) = P(PROTON but not TM) + P(TM but not PROTON) + P(PROTON and TM)

W h a t is th e

p rob ab ility th a t a t lea st

on 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 ean 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 Probability is the probability of a particular event occurring, given that another event has occurred.

5- 30

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 Rule of

Multiplication is used to find the joint probability that two events will occur.

5- 31

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- 32

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 INSANIAH University college collected the following information about undergraduate students in her college:

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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- 34

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 Diagram is useful for portraying conditional and joint probabilities. It is particularly useful for analyzing business decisions involving several stages.

5- 35

Example 8 continued

R1

B1

R2

B2

R2

B2

7/12

5/12

6/11

5/11

7/11

4/11

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Bayes’ Theorem

)/()()/()()/()(

)|(2211

111 ABPAPABPAP

ABPAPBAP

Bayes’ Theorem is a method for revising a probability given additional information.

It is computed using the following formula:

5- 37

Example 9

Duff Cola Company recently received several complaints that their bottles are under-filled. A complaint was received today but the production manager is unable to identify which of the two Springfield plants (A or B) filled this bottle. What is the probability that the under-filled bottle came from plant A?

5- 38

Example 9 continued

The following table summarizes the Duff production experience.

 % of total production

% of underfilled

bottle

A 5.5 3.0

B 4.5 4.0

5- 39

Example 9 continued

4783.)04(.45.)03(.55.

)03(.55.)/()()/()(

)/()()/(

BUPBPAUPAPAUPAP

UAP

The likelihood the bottle was filled in Plant A is reduced from .55 to .4783.

5- 40

Some Principles of Counting

Example 10: Dr. Fauzi has 10 shirts and 8 ties. How many shirt and tie outfits does he have?

The Multiplication Formula 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

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Some Principles of Counting

)!(!rn

nP rn

A Permutation is any arrangement of r objects selected from n possible objects.

Note: The order of arrangement is important in permutations.

5- 42

Some Principles of Counting

)!(!!

rnrnCrn

A Combination is the number of ways to choose r objects from a group of n objects without regard to order.

5- 43

792)!512(!5

!12512

C

There are 12 players on the Muar High School basketball team. Coach Ahmad must pick 5 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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040,95)!512(

!12512

P

Suppose that in addition to selecting the group, he must also rank each of the players in that starting lineup according to their ability (order matters).

5- 45