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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