Statistical techniques in business economics CHapter 05

Subjective ProbabilityExamples of subjective probability are: estimating the probability the Washington Redskins will win the Super Bowl this year.. The Complement Rule is used to determ

A Survey of Probability Concepts

GOALS

When you have completed this chapter, you

will be able to:

Understand the terms: experiment, event, outcome,

permutations, and combinations

Goals

A Survey of Probability Concepts

GOALS

When you have completed this chapter, you

will be able to:

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Definitions continued

There are three definitions of probability: classical,

empirical, and subjective

of times the event happens is

divided by the number of

observations

Subjective

probability is based on

whatever information is available

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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

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

Collectively Exhaustive Events

Events are Collectively Exhaustive

if at least one of the events must occur

when an experiment is conducted.

Example 2

155

0 1200

186 )

To find the probability a selected student earned an A:

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

Basic Rules of Probability

their respective probabilities

New England Commuter Airways recently supplied

the following information on their commuter flights

from Boston to New York:

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.

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

The Complement Rule continued

A

~ A

A Venn Diagram illustrating the complement rule

would appear as:

Example 4

If D is the event that a

flight is canceled, then

P(D) = 25/1000 = 025.

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.

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

The General Rule of Addition

If A and B are two

events that are not

The General Rule of

Addition

A and B A

B The Venn Diagram illustrates this rule:

EXAMPLE 5

Stereo 320

Both 100

TV 175

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.

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?

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.

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.

Example 6

5-year stock prices

0 5 10 15 20 25 30 35 40 45

P(IBM and GE) = (.5)(.7) = 35

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?

Example 6 continued

P(at least one)

= P(IBM but not GE) + P(GE but not IBM) + P(IBM and GE)

= 85

A Conditional Probability is the

probability of a particular event occurring,

given that another event has occurred.

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

General Multiplication Rule

The joint probability,

P(A and B), is given by the

Example 7 continued

P(A|F) = P(A and F)/P(F)

= [110/1000]/[400/1000] = 275

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?

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 Tree Diagram is

5/12

6/11

5/11 7/11 4/11

Bayes’ Theorem

) /

( ) (

) /

( ) (

) /

( )

( )

|

(

2 2

1 1

1

1

1

A B

P A

P A

B P A

P

A B

P A

P B

A

P





Bayes’ Theorem is a method

for revising a probability

given additional information

It is computed using the following formula:

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?

% of underfilled bottle

Example 9 continued

4783

.)

04(

45

)03(

55

)03(

55

)/

()

()

/(

)(

)/

()

()

/(

P B

P A

U P

A P

A U

P A

P U

A

P

The likelihood the bottle was filled in

Plant A is reduced from 55 to 4783

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

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

Some Principles of Counting

)!

(

!

r n

A Permutation is any arrangement of r

objects selected from n possible objects.

Note: The order of arrangement is important in

permutations.

Some Principles of Counting

Example 11

792 )!

5 12

(

! 5

must pick five

players among the

twelve on the team to

comprise the starting

lineup How many

different groups are

does not matter.)

Example 11 continued

040 ,

95 )!

5 12