Chapter04 - Basic Probability

Statistics for Managers

Using Microsoft® Excel

5th Edition

Chapter 4

Learning Objectives

In this chapter, you will learn:



_{Basic probability concepts}



_{Conditional probability}



_{To use Bayes’ Theorem to revise}

Definitions



_Probability

_{: the chance that an uncertain}

event will occur (always between 0 and 1)



_Event

_{: Each possible type of occurrence or}

outcome



_{Simple Event}

_{: an event that can be described}

by a single characteristic



_{Sample Space}

_{: the collection of all possible}

Types of Probability

There are three approaches to assessing the probability of an

uncertain event:

1.

a priori

classical probability

: the probability of an event is based

on prior knowledge of the process involved.

2.

empirical classical probability

: the probability of an event is

based on observed data.

Calculating Probability

1.

a priori

classical probability

2.

empirical classical probability

These equations

assume all outcomes are equally likely.

outcomes

possible

of

number

total

number

Probabilit





T

X

observed

outcomes

of

number

total

observed

outcomes

favorable

of

number

Occurrence

of

y

Example of

a priori

classical probability

Find the probability of selecting a face card (Jack,

Queen, or King) from a standard deck of 52 cards.

cards

of

number

total

cards

face

of

number

Probabilit

_

_

Example of

empirical

classical probability

Taking

Stats

Not Taking

Stats

Total

Male

84

145

229

Female

76

134

210

Total

160

279

439

Find the probability of selecting a male taking statistics

from the population described in the following table:

191

number

total

stats

taking

males

of

Examples of Sample Space

The Sample Space is the collection of all possible

events

ex. All 6 faces of a die:

ex. All 52 cards in a deck of cards

Events in Sample Space



_{Simple event}



_{An outcome from a sample space with one}

characteristic



_{ex. A red card from a deck of cards}



_{Complement of an event A (denoted A}

/

)



_{All outcomes that are not part of event A}



_{ex. All cards that are not diamonds}



_{Joint event}

Visualizing Events in

Sample Space



_{Contingency Tables:}



_{Tree Diagrams:}

Ace

Not

Ace

Total

Black 2

24

26

Red

2

24

26

Total

4

48

52

Full Deck

of 52 Cards

Red Card

Black

Card

Not an Ace

Ace

Ace

Not an Ace

Sample

Space

2

24

Definitions

Simple vs. Joint Probability



_{Simple (Marginal) Probability refers to the}

probability of a simple event.



_{ex. P(King)}



_{Joint Probability refers to the probability of}

an occurrence of two or more events.

Definitions

Mutually Exclusive Events



_{Mutually exclusive events}

_{are events that cannot occur}

together (simultaneously).



_example:



_{A = queen of diamonds; B = queen of clubs}



_{Events A and B are mutually exclusive if only one card is}

selected



_example:



_{B = having a boy; G = having a girl}



_{Events B and G are mutually exclusive if only one child is}

Definitions

Collectively Exhaustive Events



_{Collectively exhaustive events}



_{One of the events must occur}



_{The set of events covers the entire sample space}



_example:



_{A = aces; B = black cards; C = diamonds; D = hearts}



_{Events A, B, C and D are}

_{collectively exhaustive}

_{(but not}

mutually exclusive – a selected ace may also be a heart)



_{Events B, C and D are}

_{collectively exhaustive}

_{and also}

Computing Joint and

Marginal Probabilities



_{The probability of a joint event, A and B:}



_{Computing a marginal (or simple) probability:}



_{Where B}

₁

_{, B}

₂

_{, …, B}

_k

_{are k mutually exclusive and}

collectively exhaustive events

Example:

Joint Probability

P(Red and Ace)

52

Ace

Total

Black

2

24

26

Red

2

24

26

Example:

Marginal (Simple) Probability

P(Ace)

Ace

Not Ace

Total

Black

2

24

26

Red

2

24

26

Joint Probability Using a

Contingency Table

P(A

1

and B

2

)

P(A

1

)

Total

Event

P(A

₂

and B

₁

)

P(A

₁

and B

₁

)

Event

Total

1

Joint Probabilities

Marginal (Simple) Probabilities

A

₁

A

₂

B

₁

B

₂

P(B

₁

)

P(B

₂

)

Probability

Summary So Far



_{Probability is the numerical measure of the}

likelihood that an event will occur.



_{The probability of any event must be}

between 0 and 1, inclusively



_{0 ≤ P(A) ≤ 1 for any event A.}



_{The sum of the probabilities of all mutually}

exclusive and collectively exhaustive

events is 1.



_{P(A) + P(B) + P(C) = 1}



_{A, B, and C are mutually exclusive and}

collectively exhaustive

Certain

General Addition Rule

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

General Addition Rule:

If A and B are mutually exclusive, then

P(A and B) = 0, so the rule can be simplified:

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

General Addition Rule

Example

Taking Stats Not Taking Stats

Total

Male

84

145

229

Female

76

134

210

Total

160

279

439

Find the probability of selecting a male or a statistics student

from the population described in the following table:

P(Male or Stat) = P(M) + P(S) – P(M AND S)

Conditional Probability



_{A conditional probability is the probability of one event,}

given that another event has occurred:

P(B)

The conditional

probability of A given

that B has occurred

The conditional

Computing Conditional

Probability



_{Of the cars on a used car lot, 70% have air}

conditioning (AC) and 40% have a CD

player (CD). 20% of the cars have both.



_{What is the probability that a car has a CD}

player, given that it has AC ?

Computing Conditional

Probability

CD

No CD

Total

AC

0.2

0.5

0.7

No

AC

0.2

0.1

0.3

Total

0.4

0.6

1.0

.2857

.7

.2

P(AC)

AC)

and

P(CD

AC)

|

P(CD

_

_

_

Computing Conditional

Probability: Decision Trees

Has

CD

have AC

Has A

C

Does not

have AC

Computing Conditional

Probability: Decision Trees

Has

AC

have CD

Has C

D

Does not

have CD

Statistical Independence



_{Two events are}

_independent

_{if and only if:}



_{Events A and B are independent when the}

probability of one event is not affected by the

other event

P(A)

B)

|

Multiplication Rules



_{Multiplication rule for two events A and B:}



_{If A and B are independent, then}

and the multiplication rule simplifies to:

P(B)

B)

|

P(A

B)

and

P(A

_

P(B)

P(A)

B)

and

P(A

_

P(A)

B)

|

Multiplication Rules



_{Suppose a city council is composed of 5}

democrats, 4 republicans, and 3 independents.

Find the probability of randomly selecting a

democrat followed by an independent.



_{Note that after the democrat is selected (out of 12}

people), there are only 11 people left in the

sample space.

.114

5/44

2)

(3/11)(5/1

P(D)

D)

|

P(I

D)

and

Marginal Probability Using

Multiplication Rules

)



_{Marginal probability for event A:}



_{Where B}

Bayes’ Theorem



_{Bayes’ Theorem is used to revise previously}

calculated probabilities based on new

information.



_{Developed by Thomas Bayes in the 18}

th

Century.

Bayes’ Theorem

Bayes’ Theorem

Example



_{A drilling company has estimated a 40% chance of}

striking oil for their new well.



_{A detailed test has been scheduled for more}

information. Historically, 60% of successful wells

have had detailed tests, and 20% of unsuccessful

wells have had detailed tests.



_{Given that this well has been scheduled for a}

Bayes’ Theorem

Example



_{Let S = successful well}

U = unsuccessful well



_{P(S) = .4 , P(U) = .6 (prior probabilities)}



_{Define the detailed test event as D}



_{Conditional probabilities:}



_{P(D|S) = .6 P(D|U) = .2}

Bayes’ Theorem

Apply Bayes’ Theorem:

Bayes’ Theorem

Example



_{Given the detailed test, the revised probability of a successful}

well has risen to .667 from the original estimate of 0.4.

Event

Prior

Prob.

Conditional

Prob.

_Prob.

Joint

Revised

_Prob.

S

(successful)

.4

.6

.4*.6 = .24

.24/.36 = .667

Chapter Summary

In this chapter, we have



_{Discussed basic probability concepts.}



_{Sample spaces and events, contingency tables, simple}

probability, and joint probability



_{Examined basic probability rules.}



_{General addition rule, addition rule for mutually}

exclusive events, rule for collectively exhaustive events.



_{Defined conditional probability.}



_{Statistical independence, marginal probability, decision}

trees, and the multiplication rule