Chapter 4
Probability and
Probability Models
Copyright ©2017 McGraw-Hill Education. All rights reserved.
4-1
Probability
4.1 Probability, Sample Spaces, and
Probability Models
4.2 Probability and Events
4.3 Some Elementary Probability Rules
4.4 Conditional Probability and Independence
Copyright ©2017 McGraw-Hill Education. All rights reserved.
4-2
Learning Objective
4-1: Define a
probability, a
sample space, and
a probability
model.
•
•
•
•
•
4.1 Probability, Sample
Spaces, and Probability Models
An experiment is any process of observation
with an uncertain outcome
The possible outcomes for an experiment are
called the experimental outcomes
Probability is a measure of the chance that an
experimental outcome will occur when an
experiment is carried out
The sample space of an experiment is the set
of all possible experimental outcomes
The experimental outcomes in the sample
space are called sample space outcomes
Copyright ©2017 McGraw-Hill Education. All rights reserved.
4-3
Learning Objective
4-1: Define a
probability, a
sample space, and
a probability
model.
Probability Conditions
If E is an experimental outcome, then P(E)
denotes the probability that E will occur and:
Conditions
1. 0  P(E)  1 such that:
– If E can never occur, then P(E) = 0
– If E is certain to occur, then P(E) = 1
2. The probabilities of all the experimental
outcomes must sum to 1
Copyright ©2017 McGraw-Hill Education. All rights reserved.
4-4
Learning Objective
4-1: Define a
probability, a
sample space, and
a probability
model.
Assigning Probabilities to
Sample Space Outcomes
1. Classical method
• For equally likely outcomes
2. Relative frequency method
• Using the long run relative frequency
3. Subjective method
• Assessment based on experience,
expertise or intuition
Copyright ©2017 McGraw-Hill Education. All rights reserved.
4-5
Learning Objective
4-1: Define a
probability, a
sample space, and
a probability
model.
•
•
•
Probability Models
Probability model: a mathematical
representation of a random phenomenon
Random variable: a variable whose value
is numeric and is determined by the
outcome of an experiment
Probability distribution: A probability
model describing a random variable
1. Discrete probability distributions (Chapter
5)
2. Continuous probability distributions
(Chapter 6)
Copyright ©2017 McGraw-Hill Education. All rights reserved.
4-6
Learning Objective
4-1: Define a
probability, a
sample space, and
a probability
model.
•
Some Important Probability
Distributions
Discrete probability distributions
1. Binomial distribution
2. Poisson distribution
•
Continuous probability distributions
1. Normal distribution
2. Exponential distribution
Copyright ©2017 McGraw-Hill Education. All rights reserved.
4-7
Learning Objective
4-2: List the
outcomes in a
sample space and
use the list to
compute
probabilities.
•
•
•
4.2 Probability and Events
An event is a set of sample space
outcomes
The probability of an event is the sum of
the probabilities of the sample space
outcomes
If all outcomes equally likely, the
probability of an event is just the ratio of
the number of outcomes that correspond to
the event divided by the total number of
outcomes
Copyright ©2017 McGraw-Hill Education. All rights reserved.
4-8
Learning Objective
4-2: List the
outcomes in a
sample space and
use the list to
compute
probabilities.
•
•
A newly married couple plans to have two
children
Would like to know all possible outcomes
– BB
•
Example 4.1 Classical
Method
BG
GB
GG
Want to know probabilities
– Assuming all equal
– P(BB) = P(BG) = P(GB) = P(GG) = ¼
Copyright ©2017 McGraw-Hill Education. All rights reserved.
4-9
Learning Objective
4-2: List the
outcomes in a
sample space and
use the list to
compute
probabilities.
•
•
A company is choosing a new CEO
There are four candidates
–
–
–
–
•
Example 4.3 Subjective
Andy (A)
Chung (C)
Hill (H)
Rankin (R)
An industry analysts feels the probabilities are:
–
–
–
–
P(A) = 0.1
P(C) = 0.2
P(H) = 0.5
P(R) = 0.2
Copyright ©2017 McGraw-Hill Education. All rights reserved.
4-10
Learning objective
4-3: Use
elementary
profitability rules
to compute
probabilities.
4.3 Some Elementary
Probability Rules
1. Complement
2. Union
3. Intersection
4. Addition
5. Conditional probability
6. Multiplication
Copyright ©2017 McGraw-Hill Education. All rights reserved.
4-11
Learning objective
4-3: Use
elementary
profitability rules
to compute
probabilities.
Complement

The complement A
of an event A is the set of all sample space
outcomes not in A
P( A)  1 – P(A)
Copyright ©2017 McGraw-Hill Education. All rights reserved.
4-12
Learning objective
4-3: Use
elementary
profitability rules
to compute
probabilities.
Figure 4.3 Complement
Copyright ©2017 McGraw-Hill Education. All rights reserved.
4-13
Learning objective
4-3: Use
elementary
profitability rules
to compute
probabilities.
•
The union of A and B are elementary
events that belong to either A or B or both
–
•
Union and Intersection
Written as A  B
The intersection of A and B are elementary
events that belong to both A and B
–
Written as A ∩ B
Copyright ©2017 McGraw-Hill Education. All rights reserved.
4-14
Learning objective
4-3: Use
elementary
profitability rules
to compute
probabilities.
Example 4.4 Some Elementary
Probability Rules (1 of 2)
a) The event A is the shaded region
b) The event B is the shaded region
Copyright ©2017 McGraw-Hill Education. All rights reserved.
4-15
Learning objective
4-3: Use
elementary
profitability rules
to compute
probabilities.
Example 4.4 Some Elementary
Probability Rules (2 of 2)
c) The event A ∩ B is the shaded region
d) The event A  B is the shaded region
Copyright ©2017 McGraw-Hill Education. All rights reserved.
4-16
Learning objective
4-3: Use
elementary
profitability rules
to compute
probabilities.
•
•
Mutually Exclusive
A and B are mutually exclusive if they have
no sample space outcomes in common
In other words: P(A∩B) = 0
Copyright ©2017 McGraw-Hill Education. All rights reserved.
4-17
Learning objective
4-3: Use
elementary
profitability rules
to compute
probabilities
Example 4.5 Mutually
Exclusive
Copyright ©2017 McGraw-Hill Education. All rights reserved.
4-18
Learning objective
4-3: Use
elementary
profitability rules
to compute
probabilities.
•
•
The Addition Rule
If A and B are mutually exclusive, then the
probability that A or B (the union of A and
B) will occur is P(A  B) = P(A) + P(B)
If A and B are not mutually exclusive: P(A
 B) = P(A) + P(B) – P(A ∩ B) where
P(A∩B) is the joint probability of A and B
both occurring together
Copyright ©2017 McGraw-Hill Education. All rights reserved.
4-19
Learning objective
4-4: Compute
conditional
probabilities and
assess
independence.
•
The probability of an event A, given that
the event B has occurred, is called the
conditional probability of A given B
–
•
4.4 Conditional Probability
and Independence
Denoted as P(A|B)
Further, P(A|B) = P(A ∩ B) / P(B) – P(B) ≠
0
Copyright ©2017 McGraw-Hill Education. All rights reserved.
4-20
Learning objective
4-4: Compute
conditional
probabilities and
assess
independence.
The General Multiplication
Rule
There
• Given
• P(A ∩
• P(A ∩
•
are two ways to calculate P(A∩B)
any two events A and B
B) = P(A) P(B|A)
B) = P(B) P(A|B)
Copyright ©2017 McGraw-Hill Education. All rights reserved.
4-21
Learning objective
4-4: Compute
conditional
probabilities and
assess
independence.
Interpretation
Restrict sample space to just event B
• The conditional probability P(A|B) is the
chance of event A occurring in this new
sample space
• In other words, if B occurred, then what
is the chance of A occurring
•
Copyright ©2017 McGraw-Hill Education. All rights reserved.
4-22
Learning objective
4-4: Compute
conditional
probabilities and
assess
independence.
Independence of Events
Two events A and B are said to be
independent if and only if: P(A|B) =
P(A)
• This is equivalent to P(B|A) = P(B)
• Assumes P(A) and P(B) greater than
zero
•
Copyright ©2017 McGraw-Hill Education. All rights reserved.
4-23
Learning objective
4-4: Compute
conditional
probabilities and
assess
independence.
•
The Multiplication Rule
The joint probability that A and B (the
intersection of A and B) will occur is
PA  B   PA   PB | A   PB   PA |B 
•
If A and B are independent, then the
probability that A and B will occur is:
PA  B  PA PB  PBPA 
Copyright ©2017 McGraw-Hill Education. All rights reserved.
4-24
Learning objective
4-4: Compute
conditional
probabilities and
assess
independence.
•
Table 4.3 Contingency
Tables
A Contingency Table Summarizing Crystal’s
Cable Television and Internet Penetration
(Figures in Millions of Cable Passings)
Events
Has cable
Does Not Have
Total
Internet Service , B Internet Service , B
Has Cable Television Service 6.5
5.9
12.4
Does Not Have Cable
3.3
11.7
15.0
Television Service , A
Total
9.8
17.6
27.4
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4-25