QBM117Business Statistics

Probability

Conditional Probability

Objectives

• To calculate probabilities from contingency tables

• To develop an understanding of conditional probability

• To introduce the multiplication rule

• To differentiate between dependent and independent events

Contingency Tables

• There are several ways in which a sample space can be viewed.

• One way involves assigning the appropriate events to a contingency table.

Example 1

The police force consists of 1200 officers, of which 960 are males ad 240 are females. Over the past two years, 324 officers on the police force have been awarded promotions, of which 288 are males and 36 are females.

Present this information in a contingency table.

SexMale Female Total

Promoted 288 36 324

Notpromoted

672 204 876PromotionalStatus

Total 960 240 1200

Calculating Probabilities from a Contingency Table

• We can calculate simple and joint probabilities from contingency tables.

• Simple probability is also called marginal probability, as the total number of members in an event can be obtained from the appropriate margin of the contingency table.

SexMale Female Total

Promoted 288 36 324

Notpromoted

672 204 876PromotionalStatus

Total 960 240 1200

Example 1 revisited

What is the probability that a randomly selected officer is a man?

What is the probability that a randomly selected officer is a man?

M = event that an officer is a male

F = event that an officer is a female

Note that F is the complement of M and hence

P(F) = 1-P(M)

960( )

12000.80

P M

240( )

12000.20

P F

SexMale Female Total

Promoted 288 36 324

Notpromoted

672 204 876PromotionalStatus

Total 960 240 1200

What is the probability that a randomly selected officer is promoted?

What is the probability that a randomly selected officer is not promoted?

A = event that an officer is promoted

A = event that an officer is not promoted

Note that A is the complement of A and hence

P(A) = 1-P(A)

324( )

12000.27

P A

876( )

12000.73

P A

SexMale Female Total

Promoted 288 36 324

Notpromoted

672 204 876PromotionalStatus

Total 960 240 1200

What is the probability that a randomly selected officer is a man and is promoted?

What is the probability that a randomly selected officer is a man and is not promoted?

The probability that a randomly selected officer is male and is promoted:

The probability that a randomly selected officer is male and is not promoted:

288( )

12000.24

and P M A

672( )

12000.56

and P M A

The probability that a randomly selected officer is female and is promoted:

The probability that a randomly selected officer is female and is not promoted:

36( )

12000.03

and P F A

204( )

12000.17

and P F A

Conditional Probability

• Often the probability of an event is influenced by whether of not a related event has occurred.

• The probability of such an event is called a conditional probability.

• We want to know the probability of the event given the condition that a related event has occurred.

• The probability of event A occurring, given that event B has already occurred is

• The probability of event B occurring, given that event A has already occurred is

( and )( | )

( )

P A BP A B

P B

( and A)( | )

( )

P BP B A

P A

Example 2

Why are some mutual fund managers more successful than others? One possible factor is where the manager earned his or her MBA.

17% of mutual funds outperform the market. 40% of mutual funds are managed by a graduated from a top 20 MBA.11% of mutual funds outperform the market and are managed by a graduated from a top 20 MBA program.

What is the probability that a mutual fund outperforms the market given that the manager graduated from a top 20 program.

Let A be the event that a mutual fund outperforms the market.

Let B be the event that a mutual fund is managed by a graduate from a top 20 MBA program.

P(A) = 0.17

P(B) = 0.40

P(A and B) = 0.11

Probability that a mutual fund outperforms the market given that the manager graduated from a top 20 program:

( and )( | )

( )

0.11

0.40 0.275

P A BP A B

P B

Calculating Conditional Probabilities from a Contingency Table

• We can calculate conditional probabilities from contingency tables.

• This will be demonstrated by example.

SexMale Female Total

Promoted 288 36 324

Notpromoted

672 204 876PromotionalStatus

Total 960 240 1200

Example 1 revisited

What is the probability that an officer is promoted given that the officer is male?

What is the probability that an officer is promoted given that the officer is female?

We are no longer interested in the promotional status of all 1200 officers.

We are only interested in the promotional status of the 960 male officers.

288 of the 960 male officers were promoted.

Hence the probability that an officer is promoted given that the officer is male is

288( | )

9600.30

P A M

We are no longer interested in the promotional status of all 1200 officers.

We are only interested in the promotional status of the 240 female officers.

36 of the 240 male officers were promoted.

Hence the probability that an officer is promoted given that the officer is female is

36( | )

2400.15

P A F

Example 3(Example 4.3 from text)

Keep Kool Pty Ltd manufactures window air-conditioners in both a deluxe model and a standard model. An auditor engaged in a compliance audit of the firm is validating the sales account for April. She has collected 200 invoices for the month, some of which were sent to wholesalers and the remainder to retailers. Of the 140 retail invoices, 28 are for the standard model. Of the wholesale invoices, 24 are for the standard model.

Display this information in a contingency table.

Invoice

Wholesale Retail Total

Model

Deluxe 36 112 148

Standard 24 28 52

Total 60 140 200

If the auditor selects one invoice at random, find the following probabilities:

The invoice selected is for the deluxe model.

The invoice selected is a wholesale invoice for the deluxe model.

The invoice selected is either a wholesale invoice or and invoice for the standard model.

The events of interest are as follows:

W: wholesale invoice is selected

W: retail invoice is selected

D: invoice for deluxe model is selected

D: invoice for standard model is selected

The probability that the invoice selected is for the deluxe model:

The probability that the invoice selected is a wholesale invoice for the deluxe model:

148( )

2000.74

P D

36( and )

2000.18

P W D

The probability that the invoice selected is either a wholesale invoice or and invoice for the standard model:

36 24 28( or )

20088

2000.44

P W D

Suppose that we are told what the invoice selected by the auditor is a wholesale invoice. What is the probability that this invoice is for the deluxe model?

36( | )

60 0.60

P D W

Independent Events

• Two events are independent if neither event is affected by the occurrence of the other.

• If the occurrence of one event changes the probability of the occurrence of the others event, then the two events are dependent.

• If the occurrence of one event does not change the probability of the occurrence of the others event, then the two events are independent.

Two events A and B are independent if

the occurrence of event A does not change the probability of event B occurring

OR

the occurrence of event B does not chance the probability of event A occurring

( | ) ( )P A B P A

( | ) ( )P B A P B

Example 1 revisited

The police force consists of 1200 officers, of which 960 are males ad 240 are females. Over the past two years, 324 officers on the police force have been awarded promotions, of which 288 are males and 36 are females.

Is promotional status dependent on sex of the officer?

The probability that an officer is promoted is

The probability that an officer is promoted given that the officer is male is

The probability that an officer is promoted given that the officer is female is

( ) 0.27P A

( | ) 0.30P A M

( | ) 0.15P A F

This shows that the probability of a promotion is influenced by the sex of the officer.

Hence promotional status is dependent on the sex of the officer.

( | ) 0.30 ( ) 0.27 P A M P A

( | ) 0.15 ( ) 0.27 P A F P A

Caution

• Students often think that independent events and mutually exclusive events are the same thing.

• They are not.

• Two events are mutually exclusive if they have no members in common, i.e. P(A and B)=0

• Two events are independent if the occurrence of one does not affect the occurrence of the other, i.e. P(A|B)=P(A)

Example 2 revisited

Why are some mutual fund managers more successful than others? One possible factor is where the manager earned his or her MBA.

17% of mutual funds outperform the market. 40% of mutual funds are managed by a graduated from a top 20 MBA.11% of mutual funds outperform the market and are managed by a graduated from a top 20 MBA program.

Is whether the fund outperforms the market independent of whether the manager graduated from a top 20 MBA program?

The probability that a fund outperforms the market is

The probability that a fund outperforms the market given that the fund is managed by a graduate from a top 20 MBA program is

, hence event A is dependent on event B.

Whether the fund outperforms the market is dependent on whether the manager graduated from a top 20 MBA program

( ) 0.17P A

( | ) 0.275P A B

( | ) ( )P A B P A

Multiplication Rule

• The multiplication rule can be used to calculate the probability of the intersection of two events.

• It is a simple rearrangement of the definition of conditional probability.

( ) ( ). ( | )

( ). ( | )

and P A B P A P B A

P B P A B

Example 4

Consider a newspaper circulation department where it is known that 84% of the newspaper’s customers subscribe to the daily edition of the paper. In addition, it is known that the probability that a customer who already holds a daily subscription also subscribes to the Sunday edition is 0.75.

What is the probability that a customer subscribes to both the Sunday and daily editions of the newspaper?

Define D as the event that a customer subscribes to the daily edition.

Define S as the event that a customer subscribes to the Sunday edition.

( ) 0.84P D

( | ) 0.75P S D

( ) ( ). ( | )

0.84 0.75

0.63

and P S D P D P S D

Multiplication Rule for Independent Events

• For the special case in which A and B are independent events

• The multiplication rule then becomes

( | ) ( )P A B P A

( ) ( ). ( ) and P A B P A P B

Example 5

A group of 200 students is randomly selected. 120 of the students are female. 140 of the students are full-time, and of these 84 are female.

Show that the event of being female and independent of the event of the event of being full-time.

Calculate the probability that a student is full-time and female.

Define A as the event that a student is female.

Define B as the event that a student is full-time.

Hence the event of being female is independent of the event of being full-time.

120( )

2000.60

P A

84( | )

1400.60

P A B

( | ) ( )P A B P A

The probability that a student is full-time and female is

140( )

2000.70

P B

( ) ( ). ( ) and P A B P A P B

( ) 0.60 .70

0.42

and P A B

Reading for next lecture

• Chapter 4 sections 4.5 - 4.7

Exercises

• 4.70• 4.71• 4.73