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Introduction to ProbabilityAssigning Probabilities and Probability

Relationships

Chapter 4BA 201

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ASSIGNING PROBABILITIES

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Assigning Probabilities

Basic Requirements for Assigning Probabilities

1. The probability assigned to each experimental outcome must be between 0 and 1, inclusively.

0 < P(Ei) < 1 for all i

where:Ei is the ith experimental outcome and P(Ei) is its probability

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Assigning Probabilities

Basic Requirements for Assigning Probabilities

2. The sum of the probabilities for all experimental outcomes must equal 1.

P(E1) + P(E2) + . . . + P(En) = 1

where:n is the number of experimental outcomes

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Assigning Probabilities

Classical Method

Relative Frequency Method

Subjective Method

Assigning probabilities based on the assumption of equally likely outcomes

Assigning probabilities based on experimentation or historical data

Assigning probabilities based on judgment

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Classical Method

If an experiment has n possible outcomes, the classical method would assign a probability of 1/n to each outcome.

Experiment: Rolling a die

Sample Space: S = {1, 2, 3, 4, 5, 6}

Probabilities: Each sample point has a1/6 chance of occurring1/6+1/6+1/6+1/6+1/6+1/6 = 1

Example: Rolling a Die

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Relative Frequency Method

Number ofPolishers Rented

Numberof Days

01234

4 61810 2

Lucas Tool Rental would like to assign probabilitiesto the number of car polishers it rents each day. Office records show the following frequencies of dailyrentals for the last 40 days.

Lucas Tool Rental

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Each probability assignment is given by dividing

the frequency (number of days) by the total frequency

(total number of days).

Relative Frequency Method

4/404/40

ProbabilityNumber of

Polishers RentedNumberof Days

01234

4 61810 240

0.10 0.15 0.45 0.25 0.051.00

Example: Lucas Tool Rental

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Subjective Method

When economic conditions and a company’s circumstances change rapidly it might be inappropriate to assign probabilities based solely on historical data. We can use any data available as well as our experience and intuition, but ultimately a probability value should express our degree of belief that the experimental outcome will occur.

The best probability estimates often are obtained by combining the estimates from the classical or relative frequency approach with the subjective estimate.

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Subjective Method

An analyst made the following probability estimates.

Exper. OutcomeNet Gain or Loss Probability(10, 8)(10, -2)(5, 8)(5, -2)(0, 8)(0, -2)(-20, 8)(-20, -2)

$18,000 Gain $8,000 Gain $13,000 Gain $3,000 Gain $8,000 Gain $2,000 Loss $12,000 Loss $22,000 Loss

0.200.080.160.260.100.120.020.06

Example: Bradley Investments

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An event is a collection of sample points.

The probability of any event is equal to the sum of the probabilities of the sample points in the event.

If we can identify all the sample points of an experiment and assign a probability to each, we can compute the probability of an event.

Events and Their Probabilities

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Events and Their Probabilities

Event M = Markley Oil Profitable

M = {(10, 8), (10, -2), (5, 8), (5, -2)}

P(M) = P(10, 8) + P(10, -2) + P(5, 8) + P(5, -2)

= 0.20 + 0.08 + 0.16 + 0.26

= 0.70

Bradley Investments

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Events and Their Probabilities

Event C = Collins Mining Profitable

C = {(10, 8), (5, 8), (0, 8), (-20, 8)}

P(C) = P(10, 8) + P(5, 8) + P(0, 8) + P(-20, 8)

= 0.20 + 0.16 + 0.10 + 0.02

= 0.48

Example: Bradley Investments

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PROBABILITY RELATIONSHIPS

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Some Basic Relationships of Probability

There are some basic probability relationships that

can be used to compute the probability of an event

without knowledge of all the sample point probabilities.

Complement of an Event

Intersection of Two Events

Mutually Exclusive Events

Union of Two Events

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The complement of A is denoted by Ac.

The complement of event A is defined to be the event consisting of all sample points that are not in A.

Complement of an Event

Event A AcSampleSpace S

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The union of events A and B is denoted by A B

The union of events A and B is the event containing all sample points that are in A or B or both.

Union of Two Events

SampleSpace SEvent A Event B

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Union of Two Events

Event M = Markley Oil Profitable

Event C = Collins Mining Profitable

M C = Markley Oil Profitable or Collins Mining Profitable (or both)

M C = {(10, 8), (10, -2), (5, 8), (5, -2), (0, 8), (-20, 8)}

P(M C) = P(10, 8) + P(10, -2) + P(5, 8) + P(5, -2) + P(0, 8) + P(-20, 8)

= 0.20 + 0.08 + 0.16 + 0.26 + 0.10 + 0.02

= 0.82

Example: Bradley Investments

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The intersection of events A and B is denoted by A

The intersection of events A and B is the set of all sample points that are in both A and B.

SampleSpace SEvent A Event B

Intersection of Two Events

Intersection of A and B

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Intersection of Two Events

Event M = Markley Oil Profitable

Event C = Collins Mining Profitable

M C = Markley Oil Profitable and Collins Mining Profitable

M C = {(10, 8), (5, 8)}

P(M C) = P(10, 8) + P(5, 8)

= 0.20 + 0.16

= 0.36

Bradley Investments

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Mutually Exclusive Events

Two events are said to be mutually exclusive if the events have no sample points in common.

Two events are mutually exclusive if, when one event occurs, the other cannot occur.

SampleSpace SSampleSpace SEvent A Event B

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Mutually Exclusive Events

If events A and B are mutually exclusive, P(A B = 0.

The addition law for mutually exclusive events is:

P(A B) = P(A) + P(B)

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PRACTICE PROBABILITY RELATIONSHIPS

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Practice

Based on the Venn diagram, describe the following:

A B

C

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Practice

Out of forty students, 14 are taking English and 29 are taking Chemistry. Five students are in both classes.

Draw a Venn Diagram depicting this relationship.

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Practice

Students are either undergraduate students or graduate students.

Draw a Venn Diagram depicting this relationship.

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Scenario

Outcome Probability

O1 0.10

O2 0.30

O3 0.05

O4 0.15

O5 0.20

O6 0.05

O7 0.10

O8 0.05

Event Outcomes

E1 O1, O3

E2 O1, O4, O5, O6

E3 O2

E4 O7, O8

E5 O4, O5, O7

E6 O1, O7

A statistical experiment has the following outcomes, along with their probabilities and the following events, with the corresponding outcomes.

Outcomes Events

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Probabilities of Events

What is the probability of E1 [P(E1)]?

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Probabilities of Events

What is the probability of E2 [P(E2)]?

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Probabilities of Events

What is the probability of E3 [P(E3)]?

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Probabilities of Events

What outcomes are in the complement of E2 [E2c]?

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Probabilities of Events

What outcomes are in the union of E3 and E4? (E3 ⋃ E4)? What is the probability of E3 ⋃ E4 [P(E3 ⋃ E4)]?

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Probabilities of Events

What outcomes are in the intersection of E2 and E5 (E2 ⋂ E5)? What is the probability of E2 ⋂ E5 [P(E2 ⋂ E5)]?

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Probabilities of Events

Based on the available information, are E2 and E4 mutually exclusive? What is the probability of E2 ⋂ E4 [P(E2 ⋂ E4)]? What is the probability of E2 ⋃ E4 [P(E2 ⋃ E4)]?

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