Chapter 4

Introduction

We are always faced with decisions and

uncertain outcomes

◦ How likely is it that the project will be

finished on time?

◦ What is the chance that a new investment will

be profitable?

◦ What is the likelihood that the Professor will

ask a question I did not study for?

Careful and precise

Probability as a Numerical Measure

of the Likelihood of Occurrence

0 1 .5

Increasing Likelihood of Occurrence

Probability:

The event

is very

________

to occur.

The occurrence

of the event is

just as likely as

it is unlikely.

The event

is almost

_________

to occur.

Experiments and Sample Spaces

Experiment: Any process that generates

well-defined outcomes

Sample Space: The set of all experimental

outcomes for an experiment

Sample Point: A particular experimental

outcome

Experiment Outcomes

Toss a coin Head, Tail

Conduct a Sales Call Purchase, No Purchase

Play a football game Win, lose, tie

Take a class A, B, C, D, F

Select product for inspection Defective, Not Defective

Example: Bradley Investments

Bradley has invested in two stocks, Markley Oil and Collins Mining. Bradley has determined that the possible outcomes of these investments three months from now are as follows.

Investment Gain or Loss

in 3 Months (in $000)

Markley Oil Collins Mining

10

5

0

-20

8

-2

Learn How to Count

Experiment: Flip a coin.

◦ How many outcomes?

Experiment: Flip a coin twice

◦ How many outcomes?

Use Tree Diagram

Counting Rule 1: If an experiment consists of a

sequence of k steps in which there are n1 possible

results for the first step, n2 possible results for the

second step, and so on, then the total number of

experimental outcomes is given by (n1)(n2) . . . (nk).

Bradley Investments can be viewed as a

two-step experiment. It involves two stocks, each with a set of experimental outcomes.

Draw a tree diagram to represent the Experimental outcomes

Markley Oil: n1 = 4

Collins Mining: n2 = 2

A Counting Rule for

Multiple-Step Experiments

Tree Diagram

Markley Oil (Stage 1)

Collins Mining (Stage 2)

Experimental Outcomes

Examples

Consider a special license plate that only has three

numbers. How many different possible license plates

are there?

______________

Delta uses a 6 letter based system for confirmation

codes. How many different combination codes are

possible with this system?

_____________________

Consider a typical license plate that uses three numbers

and three letters. How many different possible license

plates are there under this system?

Combinations and Permutations In English, we use the word „combination‟

loosely, without thinking of the order of

things

◦ Salad is a combination of lettuce, mushrooms,

and cucumbers.

◦ The combination to my locker was 5-2-7.

7-2-5 would not work!

Getting Precise

◦ Combination – Order ______ matter

◦ Permutation – Order ________ matter

Should we call this a “Permutation Lock”?

Reminder: Permutation …Position

Counting Rule 2: Combinations

n objects taken from a large set of N

objects

Objects are taken without replacement

Order does not matter.

CN

n

N

n N nnN

!

!( )!

where: N! = N(N - 1)(N - 2) . . . (2)(1)

n! = n(n - 1)(n - 2) . . . (2)(1)

0! = 1

Combination Example

An inspector randomly selects two of five parts

to test for defects. How many combinations of

two parts can be selected.

Notice that we are selecting a small set from a

larger set.

Notice that order does not matter

This is a ___________

_____________

_____________

_____________

Counting Rule 3: Permutations

n objects are to be selected from a set of

N objects and order IS important

Number of Permutations of N Objects Taken n at a Time

where: N! = N(N 1)(N 2) . . . (2)(1)

n! = n(n 1)(n 2) . . . (2)(1)

0! = 1

P nN

n

N

N nnN

!

!

( )!

•An experiment results in MORE permutations than combinations

•Every combination of n objects can be ordered in n! different ways.

Assigning Probabilities

Basic Requirements

1. The probability assigned to each experimental

outcome must be between 0 and 1

Formally: Let Ei denote the ith experimental outcome and P(Ei) its probability, then 0 ≤ P(Ei) ≤ 1 for all i

2. The sum of the probabilities must equal 1.0

Formally: For n experimental outcomes, P(E1) + P(E2) + … + P(En) = 1

Assigning Probabilities - Methods

Classical Method

◦ Assigning Probabilities based on the

assumption of Equally likely outcomes

Example

◦ Consider _____________. What probability

should we assign to each outcome?

◦ Does your method satisfy the two basic

requirements?

Assigning Probabilities - Methods Relative Frequency Method

◦ Assigning Probabilities based on experimentation

or historical data

◦ Example: Lucas Tool Rental would like to assign

probabilities to the number of car polishers it rents each

day. Office records show the following frequencies of

daily rentals for the last 40 days.

Number of Days

0 1 2 3 4

4 6 18 10 2

Number of Polishers Rented

Assigning Probabilities - Methods Relative Frequency Method

◦ Each probability assignment is given by dividing

the frequency (number of days) by the total

frequency (total number of days).

Probability Number of Polishers Rented

Number of Days

0 1 2 3 4

4 6 18 10 2 40

.10 .15 .45 .25 .05 1.00

Assigning Probabilities - Methods

Subjective Method

◦ Based on degree of belief that the

experimental outcome will occur.

◦ Use data as well as experience, intuition, but

ultimately it is our belief that matters here.

Assigning Probabilities - Methods

Subjective Method

◦ Recall Bradley‟s investment in to Oil and

Mines

Exper. Outcome Net 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

.20

.08

.16

.26

.10

.12

.02

.06

Do these satisfy the requirements for assigning probabilities?

Events and Their Probabilities Up to now: Probabilities of single sample points.

Next: Probabilities of Collections of Sample Points

Event: Collection of Sample Points

Probability of any event: Equal to the sum of the

probabilities of the sample points in the event.

If we can:

1. Identify all sample points

2. Assign probability to each sample point

Then we can compute the probability of an Event

Events and Their Probabilities

Event M = Markley Oil Profitable

(First position in the pairs below)

List the sample points in event M

What is P(M), the probability of event M? Exper. Outcome

(10, 8)

(10, 2)

(5, 8)

(5, 2)

(0, 8)

(0, 2)

(20, 8)

(20, 2)

Probability

.20

.08

.16

.26

.10

.12

.02

.06

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

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

= _________________

= _______________

Events and Their Probabilities

Event C = Collins Mining Profitable

(Second position in the pairs below)

List the sample points in event C

What is P(C), the probability of event C? Exper. Outcome

(10, 8)

(10, 2)

(5, 8)

(5, 2)

(0, 8)

(0, 2)

(20, 8)

(20, 2)

Probability

.20

.08

.16

.26

.10

.12

.02

.06

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

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

= ____________________

= ___________________

Some Basic Relationships of

Probability 1. Complement of an Event

2. Intersection of two events

3. Union of two events

4. Mutually Exclusive Events

To Understand these ideas, we are going

to use a Venn Diagram (Draw)

Some Basic Relationships of

Probability Complement of an Event

◦ The complement of Event A consists of all

sample points not in A

◦ The complement of A is denoted by Ac

Event A Ac Sample Space S

Venn Diagram

P(A) = 1 – P(Ac)

Some Basic Relationships of

Probability Intersection of two events

◦ The intersection of Events A and B is the set

of all sample points that are in _____ A and

B

The intersection of A and B is denoted

A B

A B

Some Basic Relationships of

Probability Union of two events

◦ Union of Events A and B is the event

containing all sample points that are in A

____B ____ both.

◦ The union of Events A and B is denoted by

A B. Read A _____ B

Sample Space S Event A Event B

Some Basic Relationships of

Probability Union of two events

◦ Addition Rule Enables us to calculate the

probability of two events:

◦ The probability that event A, or event B, or

both occurs is

P(AB) = P(A) + P(B) – P(A B)

A B A B = + –

Some Basic Relationships of

Probability Mutually Exclusive Events ◦ Two events are said to be mutually exclusive if

_____________________________________

◦ Two events are mutually exclusive if when one

event occurs, ______________________

Sample Space S Event A Event B

Some Basic Relationships of

Probability 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)

Notice, there is no need to include ____________because

________________________

A B A B =

Conditional Probability

Def: The probability of an event, given

that another event has occurred is

called conditional probability

The conditional probability of A given B is

denoted by P(A|B).

The formula is

( )

( | )( )

P A BP A B

P B

Example Consider the situation of the promotion status of male

and female officers of a major metropolitan police force

in the US. The force consists of 1200 officers, 960 men

and 420 women. Over the past two years, 324 officers

on the police force received promotions. The specific

break down is as follows.

Given someone is a man, what is the probability they

are promoted?

Given someone is a woman, what is the probability they

are promoted?

Men Women Total

Promoted 288 36 324

NotPromoted 672 204 876

Total 960 240 1200

Example Continued Process: Compute frequency information into a probability

information.

Special kind of probability information – a joint probability

distribution.

Divide each of the center table entries by the total number of

officers

Men Women Total

Promoted 288 36 324

NotPromoted 672 204 876

Total 960 240 1200

Men Women Total

Promoted .24 .03 .27

NotPromoted .56 .17 .73

Total .80 .20 1.00

•Marginal

Probabilities

because of their

location in the

margins of the

table.

•Found

_____________

_____________

__________

Example Continued

What is P(Promoted)?

_______________

What is P(Promotion | Man)?

= ______________________

= _____________

What is P(Promotion | Woman)?

= __________________________= ________________

Men Women Total

Promoted 288 36 324

NotPromoted 672 204 876

Total 960 240 1200

Men Women Total

Promoted .24 .03 .27

NotPromoted .56 .17 .73

Total .80 .20 1.00

Independent Events

Def: If the probability of Event A is not changed by the

existence of event B, we would say that events A and B

are independent

The notation for this is:

Think of police example, did gender influence

promotion?

____________________________________

If promotion was not influenced by gender, then we

could have said that promotion was independent of

gender.

P(A|B) = P(A) P(B|A) = P(B)

Multiplication Rule

Recall formula for conditional probability

We can rearrange and, viola! The

Multiplication Rule

Essentially this reminds us that if we have two

pieces of information, we can find the third.

This will come up in the homework.

( )( | )

( )

P A BP A B

P B

P(A B) = P(B)P(A|B)

Multiplication Rule – SPECIAL CASE

The Multiplication Rule

How will the Multiplication Rule change for

independent events?

Hint, independence implies

One way to test for independence is to see if

Another way is to see if

P(A B) = P(B)P(A|B)

P(A|B) = P(A)

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

P(A|B) = P(A)

Example Suppose that we have two events, A and B. P(A) = .50, P(B) = .60, and

P(A B .40.

Find P(A | B)

Find P(B | A)

Are A and B independent? Why or why not?

Your

Turn

43 Million people in the United States go without health

insurance. Sample data representative of the national

health insurance for people older than 18 are given below

a) Develop a joint probability (including marginals) for these data and use the table.

Use the table to answer the remaining questions

b) What do the marginal age probabilities tell you about the age of the use

population 18 and older?

c) What is the probability that a randomly selected individual does not have health

insurance coverage?

d) If the individual is between 18 and 34, what is the probability that the individual

does not have health insurance coverage?

e) If the individual is 35 or older, what is the probability that the individual does not

have health insurance coverage?

f) If the individual does not have health insurance, what is the probability that the

individual is in the 18 to 34 group?

Yes No

18 to 34 750 170

35 and older 950 130

Bayes‟ Theorem

Often we begin our analysis with prior beliefs.

In probability, these are called prior

probabilities

After something happens, or we get new

information and we have to revise our

beliefs/probabilities.

Given new information, we calculate new

probabilities called Posterior Probabilities

Bayes’ Theorem is the process of getting the

revised probabilities.

Bayes‟ Theorem

Strategy we will use: Use the table familiar

from Conditional Probabilities section to

find Posterior Probabilities.

New Information

Application of Bayes’ Theorem

Posterior Probabilities

Prior Probabilities

Example

Only 1 in 1000 adults is afflicted with a rare disease for

which a diagnostic test has been developed. The test is

such that when an individual actually has the disease, a

positive result will occur 99% of the time, whereas an

individual without the disease will show a positive test

result only 2% of the time. If a randomly selected

individual is tested and the result is positive, what is the

probability that the individual has the disease?

Identify Prior information: P(disease) = ?

◦ .001

Introduce new information

◦ Someone tests positive for the disease

Form Prior: Given this new information, what is the

probability they actually have the disease?