recall

• Probability

• Mutually Exclusive Events

• Independent Events

• Collectively Exhaustive Events

• Three Approaches to Probability• Classical• Empirical• Subjective

Seatwork discussion

PERSON 1 PERSON 2

YES YES

YES NO

NO YES

NO NO

Seatwork discussion

PART 1 PART 2

ACCEPTABLE ACCEPTABLE

REPAIRABLE

SCRAPPED

REPAIRABLE ACCEPTABLE

REPAIRABLE

SCRAPPED

SCRAPPED ACCEPTABLE

REPAIRABLE

SCRAPPED

Seatwork discussion

a) 6/34 = 3/17b) Empirical

Seatwork discussion

a) 2/5 or 0.4b) Empirical

Seatwork discussion

a) Empiricalb) Classicalc) Classicald) Empirical

Seatwork discussion

a) Since gender equity is being considered, the company cannot promote two people of the same gender. Since there are 6 men and 3 women, the outcomes are as follows:

b) Classical

Man 1

Woman 1

Woman 2

Woman 3

Man 2

Woman 1

Woman 2

Woman 3

Man 3

Woman 1

Woman 2

Woman 3

Man 4

Woman 1

Woman 2

Woman 3

Man 5

Woman 1

Woman 2

Woman 3

Man 6

Woman 1

Woman 2

Woman 3

Seatwork discussion

A Survey of Probability Concepts

Lesson 4.2

Taken from: http://highered.mcgraw-hill.com/sites/0073401781/student_

view0/

10

Joint Probability – Venn Diagram

JOINT PROBABILITY A probability that measures the likelihood two or more events will happen concurrently.

11

Rules for Computing Probabilities

Rules of Addition

• Special Rule of Addition - If two events A and B are mutually exclusive, the probability of one or the other event’s occurring equals the sum of their probabilities.

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

• 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)

12

Addition Rule - Example

What is the probability that a card chosen at random from a standard deck of cards will be either a king or a heart?

P(A or B) = P(A) + P(B) - P(A and B)

= 4/52 + 13/52 - 1/52

= 16/52, or .3077

Try this:

18 7 25

A B

If there are 60 scores in all, 1. Find P(A), P(B), P(A and B).2. What is P(A or B)?

14

The Complement Rule

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

P(A) + P(~A) = 1

or P(A) = 1 - P(~A).

Example:

1. The events A and B are mutually exclusive. Suppose P(A) = 0.30 and P(B) = 0.20.

• What is the probability of either A or B occurring?

• What is the probability that neither A nor B will happen?

2. A study of 200 advertising firms revealed their income after taxes:

• What is the probability an advertising firm selected at random has either an income more than $1 million?

Income after Taxes Number of Firms

Under $1 million 102

$1-20 million 61

$20 million or more 37

Seatwork:

1. The chairman of the board says, “There is a 50% chance this company will earn a profit, a 30% chance it will break even and a 20% chance it will lose money next quarter. Find P(not lose money next quarter) and P(break even or lose money).

2. If the probability that you get a grade of A in Statistics is 0.25 and the probability you get a B is 0.50, find a) P(not getting an A), b) P(getting an A or B) and c) P(getting lower than a B)

3. Find the probability that a card drawn from a standard deck is a heart or face card (K, Q, J)?

17

Special Rule of Multiplication

• The special rule of multiplication requires that two events A and B are independent.

• 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)

18

Multiplication Rule-Example

A survey by the American Automobile association (AAA) revealed 60 percent of its members made airline reservations last year. Two members are selected at random. What is the probability both made airline reservations last year?

Solution:

The probability the first member made an airline reservation last year is .60, written as P(R1) = .60

The probability that the second member selected made a reservation is also .60, so P(R2) = .60.

Since the number of AAA members is very large, you may assume that

R1 and R2 are independent.

P(R1 and R2) = P(R1)P(R2) = (.60)(.60) = .36

19

Conditional Probability

A conditional probability is the probability of a particular event occurring, given that another event has occurred.

The probability of the event A given that the event B has occurred is written P(A|B).

20

General Multiplication Rule

The general rule of multiplication is used to find the joint probability that two events will occur.

Use the general rule of multiplication to find the joint probability of two events when the events are not independent.

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 event B occurring given that A has occurred.

21

General Multiplication Rule - Example

A golfer has 12 golf shirts in his closet. Suppose 9 of these shirts are white and the others blue. He gets dressed in the dark, so he just grabs a shirt and puts it on. He plays golf two days in a row and does not do laundry.

What is the likelihood both shirts selected are white?

22

General Multiplication Rule - Example

• The event that the first shirt selected is white is W1. The probability is P(W1) = 9/12

• The event that the second shirt selected is also white is identified as W2. The conditional probability that the second shirt selected is white, given that the first shirt selected is also white, is P(W2 | W1) = 8/11.

• To determine the probability of 2 white shirts being selected we use formula: P(AB) = P(A) P(B|A)

• P(W1 and W2) = P(W1)P(W2 |W1) = (9/12)(8/11) = 0.55

exercise:

• The board of directors of Company A consists of 8 men and 4 women. A four-member search committee is to be chosen at random to conduct a nationwide search for a new company president.

• What is the probability that all four members of the search committee will be women?

• What is the probability that all four members will be men?

24

Contingency Tables

A CONTINGENCY TABLE is a table used to classify sample observations according to two or more identifiable characteristics

E.g. A survey of 150 adults classified each as to gender and the number of movies attended last month. Each respondent is classified according to two criteria—the number of movies attended and gender.

25

Contingency Tables - Example

A sample of executives were surveyed about their loyalty to their company. One of the questions was, “If you were given an offer by another company equal to or slightly better than your present position, would you remain with the company or take the other position?” The responses of the 200 executives in the survey were cross-classified with their length of service with the company.

What is the probability of randomly selecting an executive who is loyal to the company (would remain) and who has more than 10 years of service?

26

Contingency Tables - Example

Event A1 happens if a randomly selected executive will remain with the company despite an equal or slightly better offer from another company. Since there are 120 executives out of the 200 in the survey who would remain with the company

P(A1) = 120/200, or .60.

Event B4 happens if a randomly selected executive has more than 10 years of service with the company. Thus, P(B4| A1) is the conditional probability that an executive with more than 10 years of service would remain with the company. 75 of the 120 executives who would remain have more than 10 years of service, so P(B4| A1) = 75/120.

Try this

• What is the probability of selecting an executive with more than 6-10 years of service?

• What is the probability of selecting an executive who would not remain with the company given that he has 6 to 10 years of service? Who is loyal or less than 1 year service?

• What is the probability of selecting an executive who has 6 to 10 years of service and who would not remain with the company?

QUIZOpen Notes

10 minutes to review

Answer the following on a piece of paper.

1. The market research department at a company plans to survey teenagers about a newly developed soft drink. Each will be asked to compare it with his/ her favorite drink. a. What is the experiment? b. What is one possible outcome? c. What is a possible event?

2. There are 90 students who will graduate from Treston High School. Fifty of them are planning to go to college. Two students are to be picked at random to carry the flag at graduation. a. What is the probability that both students are planning to go to

college?b. What is the probability that only one plans to go to college? (hint:

Find P(student1 goes to college OR student1 does not go to college))

3. In a management trainee program, 80% of the participants are female. Ninety percent of the females attended college and 78% of the males attended college. a. What is the probability of randomly picking a female who has not

attended college when choosing at random? b. Are gender and college attendance independent? Explain. c. If there were 1000 participants in all, construct a contingency

table showing both variables.