Sherlock Holmes once observed that men are insoluble puzzles except in the aggregate, where

they become mathematical certainties.

“You can never foretell what any one man will do,” observed

Holmes, “but you can say with precision what an average number

will be up to.

Individuals vary, but percentages remain constant.

So says the statistician.”

Basic Probability & Discrete Probability Distributions

Why study Probability?

To infer something about the population based on sample

observations

We use Probability Analysis to measure the chance that something will occur.

What’s the chance

If I flip a coin it will come up heads?

50-50

If the probability of flipping a coin is 50-50, explain why when I flipped a coin, six of the tosses were heads and four of the tosses were tails?

Think of probability in the long run:

A coin that is continually flipped, will 50% of the time be heads and 50% of the time be tails

in the long run.

Probability is a proportion or fraction

whose values range between 0 and 1, inclusively.

The Impossible Event

Has no chance of occurring and has a probability of

zero.

The Certain Event

Is sure to occur and has a probability of one.

Probability Vocabulary1) Experiment2) Events3) Sample Space4) Mutually Exclusive5) Collectively Exhaustive 6) Independent Events7) Compliment8) Joint Event

Experiment

An activity for which the outcome is uncertain.

Examples of an Experiment:

• Toss a coin• Select a part for inspection• Conduct a sales call• Roll a die• Play a football game

Events

Each possible outcome of

the experiment.

Examples of an Event:• Toss a coin• Select a part for

inspection• Conduct a sales

call• Roll a die• Play a football

game

• Heads or tails• Defective or non-

defective• Purchase or no

purchase• 1,2,3,4,5,or 6• Win, lose, or tie

Sample Space

The set of ALL possible outcomes of an experiment.

Examples of Sample Spaces:• Toss a coin• Select a part for

inspection• Conduct a sales

call• Roll a die• Play a football

game

• Heads, tails• Defective,

nondefective• Purchase, no

purchase• 1,2,3,4,5,6• Win, lose, tie

Mutually Exclusive Events cannot both occur simultaneously.

Collectively Exhaustive

A set of events is collectively exhaustive if one of the events must occur.

Independent Events

If the probability of one event occurring is unaffected by the occurrence or nonoccurrence of the other event.

ComplementThe complement of Event A includes all

events that are not part of Event A.

The complement of Event A is denoted by Ā or A’.

Example: The compliment of being male is being female.

Joint Event

Has two or more characteristics. Age (Years)

<30 30-45 >45 Gender (U) (B) (O) Total

Male (M) 60 20 40 120 Female (F) 40 30 10 80

Total 100 50 50 200

Probability Vocabulary1) Experiment2) Events3) Sample Space4) Mutually Exclusive5) Collectively

Exhaustive 6) Independent Events7) Compliment8) Joint Event

Quiz

What’s the difference between Mutually Exclusive and Collectively Exhaustive?

When you estimate a probability

You are estimating the probability of an EVENT occurring.

When rolling two die, the probability of rolling an 11 (Event A) is the probability that Event A occurs.

It is written P(A)

P(A) = probability that event A occurs

With a sample space of the toss of a fair die being

S = {1, 2, 3, 4, 5, 6}

Find the probability of the following events:

1) An even number2) A number less than or

equal to 43) A number greater than

or equal to 5.

Answers

1)P(even number) = P(2) + P(4) + P(6)=1/6 + 1/6 + 1/6 = 3/6 =1/2

2)P(number ≤ 4) = P(1) + P(2) + P(3) + P(4)= 1/6 + 1/6 + 1/6 + 1/6 = 4/6 =

2/33)P(number ≥ 5) = P(5) + P(6) =

1/6 + 1/6 = 2/6 = 1/3

Approaches to Assigning Probabilities

• The Relative Frequency

• The Classical Approach

• The Subjective Approach

Classical Approach to Assigning Probability

Probability based on prior knowledge of the process involved with each outcome equally likely to occur in the long-run if the selection process is continually repeated.

Relative Frequency (Empirical) Approach to Assigning Probability

Probability of an event occurring based on observed data.

By observing an experiment n times, if Event A occurs m times of the n times, the probability that A will occur in the future is

P(A) = m /n

Example of Relative Frequency Approach

1000 students take a probability exam.

200 students score an A.

P(A) = 200/1000 = .2 or 20%

The Relative Frequency Approach assigned probabilities to the following simple events

What is the probability a student will pass the course with a C or better?

P(A) = .2P(B) = .3P(C) = .25P(D) = .15P(F) = .10

Subjective Approach to Assigning Probability

Probability based on individual’s past experience, personal opinion, analysis of situation. Useful if probability cannot be determined empirically.

We leave Base Camp; the Ascent for the Summit Begins!

From a survey of 200 purchasers of a laptop computer, a gender-age profile is

summarized below:

CLASS FREQUENCY CLASS FREQUENCY Male 120 Under 30 100 Female 80 30 -45 50 Total 200 Over 45 50 Total 200

These two categories (gender and age) can be summarized

together in a contingency or cross-tab table which allows

the viewer to see how these two categories interact

CLASS FREQUENCY CLASS FREQUENCY Male 120 Under 30 100 Female 80 30 -45 50 Total 200 Over 45 50 Total 200

Age (Years)

<30 30-45 >45 Gender (U) (B) (O) Total

Male (M) 60 20 40 120 Female (F) 40 30 10 80

Total 100 50 50 200

Marginal Probability

The probability that any one single event will occur.

Example: P(M) = 120/200 = .6 Age (Years)

<30 30-45 >45 Gender (U) (B) (O) Total

Male (M) 60 20 40 120 Female (F) 40 30 10 80

Total 100 50 50 200

What’s the probability of being under 30?What’s the probability of being female?

What’s the probability of being either under 30 or over 45?

Age (Years)

<30 30-45 >45 Gender (U) (B) (O) Total

Male (M) 60 20 40 120 Female (F) 40 30 10 80

Total 100 50 50 200

What is the complement of being male? P(MC) or P(M’)

Age (Years)

<30 30-45 >45 Gender (U) (B) (O) Total

Male (M) 60 20 40 120 Female (F) 40 30 10 80

Total 100 50 50 200

Joint Probability

The probability that both Events A and B will occur.This is written as P(A and B)

Age (Years)

<30 30-45 >45 Gender (U) (B) (O) Total

Male (M) 60 20 40 120 Female (F) 40 30 10 80

Total 100 50 50 200

What is the probability of selecting a purchaser who is female and under age 30?

P(F and U) = 40/200 = .2 or 20%

Age (Years)

<30 30-45 >45 Gender (U) (B) (O) Total

Male (M) 60 20 40 120 Female (F) 40 30 10 80

Total 100 50 50 200

Probability of A or B

The probability that either of two events will occur.

This is written as P(A or B).

Use the General Addition Rule which eliminates double-counting.

General Addition Rule

P(A or B) = P(A) + P(B) – P(A and B)

What is the probability of selecting a purchaser who is male OR under 30 years of age?

P(M or U) = P(M) + P(U) – P(M and U)=(120 + 100 – 60) / 200 = 160 / 200= .8 or 80%

Age (Years)

<30 30-45 >45 Gender (U) (B) (O) Total

Male (M) 60 20 40 120 Female (F) 40 30 10 80

Total 100 50 50 200

We can use raw data

NortheastD

SoutheastE

MidwestF

WestG

Finance A 24 10 8 14 56

Manufacturing B 30 6 22 12 70

Communication C 28 18 12 16 74

82 34 42 42 200

Or we can convert our contingency table into percentages

NortheastD

SoutheastE

MidwestF

WestG

Finance A .12 .05 .04 .07 .28

Manufacturing B .15 .03 .11 .06 .35

Communication C .14 .09 .06 .08 .37

.41 .17 .21 .21 1.00

P(Midwest) = ? P(C or D) = ? P(E or A) =?

Northeast

D

Southeast

E

MidwestF

WestG

Finance A

24 10 8 14 56

Manufacturing B

30 6 22 12 70

Communication

C28 18 12 16 74

82 34 42 42 200

Northeast

D

Southeast

E

MidwestF

WestG

Finance A

.12 .05 .04 .07 .28

Manufacturing B

.15 .03 .11 .06 .35

Communication

C.14 .09 .06 .08 .37

.41 .17 .21 .21 1.00

SolutionNorthe

astD

Southeast

E

MidwestF

WestG

Finance A

.12 .05 .04 .07 .28

Manufacturing B

.15 .03 .11 .06 .35

Communication C

.14 .09 .06 .08 .37

.41 .17 .21 .21 1.00

P(F) = .21

P(C or D) = P(C) + P(D) – P(C & D)= .37 + .41 - .14= .64 or 64%

P(E or A) =.17 + .28 - .05= .40 or 40%

Addition Rule for Mutually Exclusive Events:

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

Frequently, we need to know how two events are related.

Conditional Probability

We would like to know the probability of one event occurring given the occurrence of another related event.

Conditional Probability

The probability that Event A occurs GIVEN that Event B occurs.

P (A | B)

B is the event known to have occurred and A is the uncertain event whose probability you seek, given that Event B has occurred.

What is the probability of selecting a female purchaser given the selected individual is under 30 years of age?

P(F | U) = 40 / 100 = .4

Interpretation:There is a 40% probability of selecting a female given the

selected individual is under 30 years of age.

Age (Years)

<30 30-45 >45 Gender (U) (B) (O) Total

Male (M) 60 20 40 120 Female (F) 40 30 10 80

Total 100 50 50 200

Hypoxia Question 1:

How is P(F|U) different than the P(F)?

There is a 40% chance of selecting a female purchaser given no prior information about U. P(F)= .4

This means that being given the information that the person selected is

under 30 has no effect on the probability that a female is selected.

In other words, U has no effect on whether F occurs. Such events are

said to be INDEPENDENT

Events A and B are independent if the probability of Event A is unaffected by the occurrence or non-occurence of

Event B

Statistical Independence

• Events A and B are independent if and only if:

• P(A | B) = P(A) {assuming P(B) ≠ 0}, or• P(B | A) = P(B) {assuming P(A) ≠ 0}, or• P(A and B) = P(A) ∙ P(B).

Age (Years)

<30 30-45 >45 Gender (U) (B) (O) Total

Male (M) 60 20 40 120 Female (F) 40 30 10 80

Total 100 50 50 200

What is the probability of selecting a female purchaser given the selected individual is between 30-45 years of age?

Are the events independent?

P(F | B) = 30/50 = .6

Test for independence:P(F | B) = P(F)30/50 = 80/200

.6 ≠ .4The events are not independent.

1) Suppose we have the following joint probabilities.

A1 A2 A3 B1 .15 .20 .10 B2 .25 .25 .05

1) Compute the marginal probabilities. 2) Compute P(A2 | B2) 3) Compute P(B2 | A2) 4) Compute P(B1 | A2) 5) Compute P( A1 or A2) 6) Compute P(A2 | or B2) 7) Compute P(A3 or B1)

1) The female instructors at a large university recently lodged a complaint about the most recent round of promotions from assistant professor to associate professor. An analysis of the relationship between gender and promotion was undertaken with the joint probabilities in the following table being produced.

Promoted Not Promoted Female .03 .12 Male .17 .68

• What is the rate of promotion among female assistant professors?

• What is the rate of promotion among male assistant professors?

• Is it reasonable to accuse the university of gender bias?

1) To determine whether drinking alcoholic beverages has an effect on the bacteria that cause ulcers, researchers developed the following table of joint probabilities.

i) What proportion of people have ulcers? ii) What is the probability that a teetotaler (no alcoholic beverages) develops an ulcer? iii) What is the probability that someone who has an ulcer does not drink alcohol? iv) Are ulcers and the drinking of alcohol independent? Explain.

Number of alcoholic drinks per day

Ulcer No Ulcer

None .01 .22 One .03 .19 Two .03 .32 More than two .04 .16