© 2011 Pearson Education, Inc. Statistics for Business and Economics Chapter 3 Probability.

Statistics for Business and Economics

Chapter 3

Probability

Contents1. Events, Sample Spaces, and Probability2. Unions and Intersections3. Complementary Events4. The Additive Rule and Mutually Exclusive

Events5. Conditional Probability6. The Multiplicative Rule and Independent

Events7. Random Sampling8. Baye’s Rule

Learning Objectives

1. Develop probability as a measure of uncertainty

2. Introduce basic rules for finding probabilities

3. Use probability as a measure of reliability for an inference

Thinking Challenge

• What’s the probability of getting a head on the toss of a single fair coin? Use a scale from 0 (no way) to 1 (sure thing).

• So toss a coin twice. Do it! Did you get one head & one tail? What’s it all mean?

Many Repetitions!*

Number of Tosses

Total Heads Number of Tosses

0 25 50 75 100 125

Events, Sample Spaces,and Probability

Experiments & Sample Spaces

1. Experiment• Process of observation that leads to a single

outcome that cannot be predicted with certainty

2. Sample point• Most basic outcome of an

experiment

3. Sample space (S) • Collection of all possible outcomes

Sample Space Depends on Experimenter!

Sample Space Properties

Experiment: Observe Gender

1. Mutually Exclusive

• 2 outcomes can not occur at the same time

— Male & Female in same person

2. Collectively Exhaustive

• One outcome in sample space must occur.

— Male or Female

Visualizing Sample Space

1. Listing

S = {Head, Tail}

2. Venn Diagram

Sample Space Examples

• Toss a Coin, Note Face {Head, Tail}• Toss 2 Coins, Note Faces {HH, HT, TH, TT}• Select 1 Card, Note Kind {2♥, 2♠, ..., A♦} (52)• Select 1 Card, Note Color {Red, Black} • Play a Football Game {Win, Lose, Tie}• Inspect a Part, Note Quality {Defective, Good}• Observe Gender {Male, Female}

Experiment Sample Space

Events

1. Specific collection of sample points

2. Simple Event

• Contains only one sample point

3. Compound Event

• Contains two or more sample points

Sample Space S = {HH, HT, TH, TT}

Venn Diagram

Outcome

Experiment: Toss 2 Coins. Note Faces.

Compound Event: At least one Tail

Event Examples

• 1 Head & 1 Tail HT, TH

• Head on 1st Coin HH, HT

• At Least 1 Head HH, HT, TH

• Heads on Both HH

Experiment: Toss 2 Coins. Note Faces.

Sample Space: HH, HT, TH, TT

Event Outcomes in Event

Probabilities

What is Probability?

1. Numerical measure of the likelihood that event will cccur

• P(Event)• P(A)• Prob(A)

2. Lies between 0 & 1

3. Sum of sample points is 1

CertainCertain

ImpossibleImpossible

Probability Rulesfor Sample Points

Let pi represent the probability of sample point i.

1. All sample point probabilities must lie between 0 and 1 (i.e., 0 ≤ pi ≤ 1).

2. The probabilities of all sample points within a sample space must sum to 1 (i.e., pi = 1).

Equally Likely Probability

P(Event) = X / T• X = Number of outcomes in the

• T = Total number of sample points in Sample Space

• Each of T sample points is equally likely

— P(sample point) = 1/T

Steps for Calculating Probability

1. Define the experiment; describe the process used to make an observation and the type of observation that will be recorded

2. List the sample points

3. Assign probabilities to the sample points

4. Determine the collection of sample points contained in the event of interest

5. Sum the sample points probabilities to get the event probability

Combinations RuleA sample of n elements is to be drawn from a set of N elements. The, the number of different samples possible

is denoted byN

⎛⎝⎜

⎞⎠⎟

and is equal to

⎛⎝⎜

⎞⎠⎟=

N!n! N−n( )!

where the factorial symbol (!) means that

n!=n n−1( ) n−2( )L 3( ) 2( ) 1( )

5!=5⋅4⋅3⋅2⋅1For example, 0! is defined to be 1.

Unions and Intersections

Compound Events

Compound events:

Composition of two or more other events.

Can be formed in two different ways.

Unions & Intersections

1. Union• Outcomes in either events A or B or both• ‘OR’ statement• Denoted by symbol (i.e., A B)

2. Intersection• Outcomes in both events A and B• ‘AND’ statement• Denoted by symbol (i.e., A B)

BlackAce

Event Union: Venn Diagram

Event Ace Black:

A, ..., A, 2, ..., K

Event Black:

2,...,

Sample Space:

2, ..., A

Event Ace:

A, A, A, A

Experiment: Draw 1 Card. Note Kind, Color & Suit.

EventAce Black:

A,..., A, 2, ..., K

Event Union: Two–Way Table

Sample Space (S):

2, ..., A

Simple Event Ace:

Simple Event Black:

2, ..., A

Experiment: Draw 1 Card. Note Kind, Color & Suit. Color

Type Red Black TotalAce Ace &

RedAce &Black

Non &Red

Non &Black

Non-Ace

Total Red Black S

Non-Ace

BlackAce

Event Intersection: Venn Diagram

Event Ace Black:

Event Black:

2,...,A

Sample Space:

2, ..., A

Event Ace:

A, A, A, A

Sample Space (S):

2, ..., A

Event Intersection: Two–Way Table

Ace Black:

Simple Event Ace:

Simple Event Black: 2, ..., A

ColorType Red Black Total

Ace Ace &Red

Ace &Black

Non &Red

Non &Black

Non-Ace

Total Red Black S

Non-Ace

Compound Event Probability

1. Numerical measure of likelihood that compound event will occur

2. Can often use two–way table• Two variables only

EventEvent B1 B2 Total

A1 P(A 1 B1) P(A1 B2) P(A1)

A2 P(A 2 B1) P(A2 B2) P(A2)

P(B1) P(B2) 1

Event Probability Using Two–Way Table

Joint Probability Marginal (Simple) Probability

Ace 2/52 2/52 4/52

Non-Ace 24/52 24/52 48/52

Total 26/52 26/52 52/52

Two–Way Table Example

Experiment: Draw 1 Card. Note Kind & Color.

P(Ace)

P(Ace Red)P(Red)

1. P(A) =

2. P(D) =

3. P(C B) =

4. P(A D) =

5. P(B D) =

Thinking Challenge

EventEvent C D Total

A 4 2 6

B 1 3 4

Total 5 5 10

What’s the Probability?

Solution*

The Probabilities Are:

1. P(A) = 6/10

2. P(D) = 5/10

3. P(C B) = 1/10

4. P(A D) = 9/10

5. P(B D) = 3/10

A 4 2 6

B 1 3 4

Total 5 5 10

Complementary Events

Complement of Event A• The event that A does not occur• All events not in A• Denote complement of A by AC

Rule of Complements

The sum of the probabilities of complementary events equals 1:

P(A) + P(AC) = 1

Complement of Event Example

Event Black:

2, 2, ..., A

Complement of Event Black,

BlackC: 2, 2, ..., A, A

Sample Space:

2, ..., A

Experiment: Draw 1 Card. Note Color.

The Additive Rule and Mutually Exclusive Events

Mutually Exclusive Events

• Events do not occur simultaneously

• A does not contain any sample points

Mutually Exclusive Events

Mutually Exclusive Events Example

Events and are Mutually Exclusive

Experiment: Draw 1 Card. Note Kind & Suit.

Outcomes in Event Heart:

2, 3, 4,

..., A

Sample Space:

2, ..., A

Event Spade:

2, 3, 4, ..., A

Additive Rule

1. Used to get compound probabilities for union of events

2. P(A OR B) = P(A B) = P(A) + P(B) – P(A B)

3. For mutually exclusive events:P(A OR B) = P(A B) = P(A) + P(B)

Additive Rule Example

P(Ace Black) = P(Ace) + P(Black) – P(Ace Black)

Ace 2 2 4

Non-Ace 24 24 48

Total 26 26 52

52 52 52 52 4 26 2 28

= + – =

Thinking Challenge

1. P(A D) =

2. P(B C) =

A 4 2 6

B 1 3 4

Total 5 5 10

Using the additive rule, what is the probability?

10 10 10 10 6 5 2 9

Solution*

Using the additive rule, the probabilities are:

P(A D) = P(A) + P(D) – P(A D)1.

2. P(B C) = P(B) + P(C) – P(B C)

10 10 10 10

4 5 1 8

= + – =

Conditional Probability

1. Event probability given that another event occurred

2. Revise original sample space to account for new information

• Eliminates certain outcomes

3. P(A | B) = P(A and B) = P(A B) P(B) P(B)

BlackAce

Conditional Probability Using Venn Diagram

Black ‘Happens’: Eliminates All Other Outcomes

Event (Ace Black)

(S)Black

Conditional Probability Using Two–Way Table

Revised Sample Space

Ace 2 2 4

Non-Ace 24 24 48

Total 26 26 52

P(Ace Black) 2 / 52 2P(Ace | Black) =

P(Black) 26 / 52 26

∩= =

Using the table then the formula, what’s the probability?

Thinking Challenge

1. P(A|D) =

2. P(C|B) =

A 4 2 6

B 1 3 4

Total 5 5 10

Solution*

Using the formula, the probabilities are:

P A D( )=P A∩B( )

P D( )=

P C B( )=P C∩B( )

P B( )=110

The Multiplicative Rule

and Independent Events

Multiplicative Rule

1. Used to get compound probabilities for intersection of events

2. P(A and B) = P(A B)= P(A) P(B|A) = P(B) P(A|B)

3. For Independent Events:P(A and B) = P(A B) = P(A) P(B)

Multiplicative Rule Example

Experiment: Draw 1 Card. Note Kind & Color. Color

Type Red Black Total

Ace 2 2 4

Non-Ace 24 24 48

Total 26 26 52

52 4 52⎛ ⎞⎛ ⎞= =⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

P(Ace Black) = P(Ace)∙P(Black | Ace)

1. Event occurrence does not affect probability of another event

• Toss 1 coin twice

2. Causality not implied

3. Tests for independence• P(A | B) = P(A)• P(B | A) = P(B)

• P(A B) = P(A) P(B)

Statistical Independence

Thinking Challenge

1. P(C B) =

2. P(B D) =

3. P(A B) =

A 4 2 6

B 1 3 4

Total 5 5 10

Using the multiplicative rule, what’s the probability?

Solution*

Using the multiplicative rule, the probabilities are:

P C ∩B( )=P C( )⋅P B C( )=510

⋅15=

P B∩D( )=P B( )⋅P D B( )=410

⋅35=

P A∩B( )=P A( )⋅P B A( )=0

Tree DiagramExperiment: Select 2 pens from 20 pens: 14 blue & 6 red. Don’t replace.

Dependent!

RR6/20

14/206/19

P(R R)=(6/20)(5/19) =3/38

P(R B)=(6/20)(14/19) =21/95

P(B R)=(14/20)(6/19) =21/95

P(B B)=(14/20)(13/19) =91/190

Random Sampling

Importance of Selection

How a sample is selected from a population is of vital importance in statistical inference because the probability of an observed sample will be used to infer the characteristics of the sampled population.

Random Sample

If n elements are selected from a population in such a way that every set of n elements in the population has an equal probability of being selected, the n elements are said to be a random sample.

Random Number Generators

Most researchers rely on random number generators to automatically generate the random sample.Random number generators are available in table form, and they are built into most statistical software packages.

Bayes’s Rule

Given k mutually exclusive and exhaustive events B1, B1, . . . Bk , such thatP(B1) + P(B2) + … + P(Bk) = 1,and an observed event A, then

P(Bi| A) =

P(Bi ∩ A)P(A)

=P(Bi )P(A|Bi )

P(B1)P(A|B1) + P(B2 )P(A|B2 ) + ...+ P(Bk)P(A|Bk)

Bayes’s Rule Example

A company manufactures MP3 players at two factories. Factory I produces 60% of the MP3 players and Factory II produces 40%. Two percent of the MP3 players produced at Factory I are defective, while 1% of Factory II’s are defective. An MP3 player is selected at random and found to be defective. What is the probability it came from Factory I?

Bayes’s Rule Example

Factory Factory IIII

Factory Factory II0 .6

0 .4 0.01

DefectiveDefective

GoodGood

P(I | D) =P( I )P(D |I )

P( I )P(D |I ) + P( II )P(D |II )=

0.6⋅0.020.6⋅0.02 + 0.4⋅0.01

Key Ideas

Probability Rules for k Sample Points,

S1, S2, S3, . . . , Sk

1. 0 ≤ P(Si) ≤ 1

2.P Si( )∑ =1

Key Ideas

Random Sample

All possible such samples have equal probability of being selected.

Key Ideas

Combinations Rule

Counting number of samples of n elements selected from N elements

⎛⎝⎜

⎞⎠⎟=

N!n! N−n( )!

=N N−1( ) N−2( )L N−n+1( )

n n−1( ) n−2( )L 2( ) 1( )

Key Ideas

Bayes’s Rule

i| A) =

P(Si )P(A|Si )P(S1)P(A|S1) + P(S2 )P(A|S2 ) + ...+ P(Sk)P(A|Sk)

© 2011 Pearson Education, Inc. Statistics for Business and Economics Chapter 3 Probability.

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