11

Statistics for Managers 5th Edition

Chapter 4Basic Probability

22

Chapter Topics

Basic probability concepts Sample spaces and events, simple

probability, joint probability

Conditional probability Statistical independence, marginal

probability

Bayes’s Theorem

33

Terminology

Experiment- Process of Observation

Outcome-Result of an Experiment Sample Space- All Possible

Outcomes of a Given Experiment Event- A Subset of a Sample

Space

44

Sample Spaces

Collection of all possible outcomes e.g.: All six faces of a die:

e.g.: All 52 cards in a deck:

55

Events

Simple event Outcome from a sample space with

one characteristic e.g.: A red card from a deck of cards

Joint event Involves two outcomes simultaneously e.g.: An ace that is also red from a

deck of cards

66

Visualizing Events

Contingency Tables

Tree Diagrams

Red 2 24 26

Black 2 24 26

Total 4 48 52

Ace Not Ace Total

Full Deck of Cards

Red Cards

Black Cards

Not an Ace

Ace

Ace

Not an Ace

77

Special Events

Impossible evente.g.: Club & diamond on one card

draw Complement of event

For event A, all events not in A Denoted as A’ e.g.: A: queen of diamonds

A’: all cards in a deck that are not queen of diamonds

Null Event

88

Contingency TableA Deck of 52 Cards

Ace Not anAce

Total

Red

Black

Total

2 24

2 24

26

26

4 48 52

Sample Space

Red Ace

99

Full Deck of Cards

Tree Diagram

Event Possibilities

Red Cards

Black Cards

Ace

Not an Ace

Ace

Not an Ace

1010

Probability

Probability is the numerical measure of the likelihood that an event will occur

Value is between 0 and 1 Sum of the probabilities of

all mutually exclusive and collective exhaustive events is 1

Certain

Impossible

.5

1

0

1111

Types of Probability

•Classical (a priori) Probability P (Jack) = 4/52

•Empirical (Relative Frequency) Probability Probability it will rain today = 60%

•Subjective Probability Probability that new product will be successful

1212

(There are 2 ways to get one 6 and the other 4)e.g. P( ) = 2/36

Computing Probabilities

The probability of an event E:

Each of the outcomes in the sample space is equally likely to occur

number of event outcomesP(E)=

total number of possible outcomes in sample space

n(E)

n(S)

1313

Probability Rules

1 0 ≤ P(E) ≤ 1 Probability of any event must be between 0

and12 P(S) = 1 ; P(Ǿ) = 0 Probability that an event in the sample space

will occur is 1; the probability that an event that is not in the sample space will occur is 0

3 P (E) = 1 – P(E) Probability that event E will not occur is 1

minus the probability that it will occur

14

Rules of Addition

4 Special Rule of AdditionP (AuB) = P(A) + P(B) if and only if A and B are mutually exclusive events

5. General Rule of AdditionP (AuB) = P(A) + P(B) – P(AnB)

1515

Rules Of Multiplication

6 Special Rule of MultiplicationP (AnB) = P(A) x P(B) if and only if A and B are statistically independent events

7 General Rule of MultiplicationP (AnB) = P(A) x P(B/A)

1616

Conditional Probability Rule

Conditional Probability RuleP(B/A) = P (AnB)/ P(A)

This is a rewrite of the formula for the general rule of multiplication.

1717

Bayes Theorem

P(B1) = probability that Bill fills prescription = .20

P(B2) = probability that Mary fills prescription = .80

P(A B1) = probability mistake Bill fills prescription = 0.10

P(A B2) = probability mistake Mary fills prescription = 0.01

What is the probability that Bill filled a prescription that contained a mistake?

1818

Bayes’s Theorem

1 1

||

| |

and

i ii

k k

i

P A B P BP B A

P A B P B P A B P B

P B A

P A

Adding up the parts of A in all the B’s

Same Event

1919

Bayes Theorem (cont.)

P(B1 A) =(.20) (.10)

(.20) (.10) + (.80) (.01)===

.02

.028.71 71%

2020

Bayes Theorem (cont.)

Bill fills prescription

Mary fills prescription

(Prior)Bi

.20

. 80

1.00

(Joint)A Bi

.020

.008

P(A) =.028

(Posterior) Bayes

.02/.028=.71

.008/.028=.29

1.00

(Conditional)

.10

.01

A Bi

2121

Chapter Summary Discussed basic probability

concepts Sample spaces and events, simple

probability, and joint probability

Defined conditional probability Statistical independence, marginal

probability

Discussed Bayes’s theorem