Statistics for Business and

Economics

Chapter 3

Probability

© 2011 Pearson Education, Inc

Contents

1.

2.

3.

4.

5.

6.

7.

8.

Events, Sample Spaces, and Probability

Unions and Intersections

Complementary Events

The Additive Rule and Mutually Exclusive

Events

Conditional Probability

The Multiplicative Rule and Independent

Events

Random Sampling

Baye’s Rule

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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

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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?

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Many Repetitions!*

Total Heads

Number of Tosses

1.00

0.75

0.50

0.25

0.00

0

25

50

75

Number of Tosses

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100

125

3.1

Events, Sample Spaces, and Probability

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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 Sample Space

Depends on

Experimenter!

3. Sample space (S)

• Collection of all possible outcomes

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Sample Space Properties

1. Mutually Exclusive

•

Experiment: Observe Gender

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

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© 1984-1994 T/Maker Co.

Visualizing

Sample Space

1.

Listing

S = {Head, Tail}

2.

Venn Diagram

H

T

S

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Sample Space Examples

•

•

•

•

•

•

•

Experiment

Sample Space

Toss a Coin, Note Face

Toss 2 Coins, Note Faces

Select 1 Card, Note Kind

Select 1 Card, Note Color

Play a Football Game

Inspect a Part, Note Quality

Observe Gender

{Head, Tail}

{HH, HT, TH, TT}

{2♥, 2♠, ..., A♦} (52)

{Red, Black}

{Win, Lose, Tie}

{Defective, Good}

{Male, Female}

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Events

1. Specific collection of sample points

2. Simple Event

• Contains only one sample point

3. Compound Event

• Contains two or more sample points

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Venn Diagram

Experiment: Toss 2 Coins. Note Faces.

Sample Space

S = {HH, HT, TH, TT}

TH

Outcome

HH

Compound

Event: At least one

Tail

HT

TT

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S

Event Examples

Experiment: Toss 2 Coins. Note Faces.

Sample Space: HH, HT, TH, TT

Event

• 1 Head & 1 Tail

• Head on 1st Coin

• At Least 1 Head

• Heads on Both

Outcomes in Event

HT, TH

HH, HT

HH, HT, TH

HH

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Probabilities

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What is Probability?

1. Numerical measure of the

1

likelihood that event will cccur • P(Event)

• P(A)

.5

• Prob(A)

Certain

2. Lies between 0 & 1

3. Sum of sample points is 1

0

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Impossible

Probability Rules for 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., Σ pi = 1).

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Equally Likely Probability

P(Event) = X / T

• X = Number of outcomes in the event • T = Total number of sample points in Sample Space

• Each of T sample points is equally likely — P(sample point) = 1/T

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© 1984-1994 T/Maker Co.

Steps for Calculating…