Copyright ©2017 McGraw-Hill Education. All rights reserved.
4-1

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.
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.
1.
2.
3.
Classical method
• For equally likely outcomes
Relative frequency method
• Using the long run relative frequency
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 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
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4-7

Learning Objective
4-2: List the outcomes in a sample space and use the list to compute probabilities.
•
•
•
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 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
– 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.
1.
2.
3.
4.
5.
6.
Complement
Union
Intersection
Addition
Conditional probability
Multiplication
Copyright ©2017 McGraw-Hill Education. All rights reserved.
4-11

Learning objective
4-3: Use elementary profitability rules to compute probabilities.
The complement
  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.
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
– Written as A  B
The intersection of A and B are elementary events that belong to both A and B
– Written as A ∩ B
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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.
•
•
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
Copyright ©2017 McGraw-Hill Education. All rights reserved.
4-18

Learning objective
4-3: Use elementary profitability rules to compute probabilities.
•
•
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
– 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.
•
•
•
• There are two ways to calculate P(A ∩ B)
Given any two events A and B
P(A ∩ B) = P(A) P(B|A)
P(A ∩ B) = P(B) P(A|B)
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4-21

Learning objective
4-4: Compute conditional probabilities and assess independence.
•
•
•
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 joint probability that A and B (the intersection of A and B) will occur is
P

A

B
   
B | A

 P
  
A | B

If A and B are independent, then the probability that A and B will occur is:
P

A  B

 P
       
Copyright ©2017 McGraw-Hill Education. All rights reserved.
4-24

Learning objective
4-4: Compute conditional probabilities and assess independence.
• A Contingency Table Summarizing Crystal’s
Cable Television and Internet Penetration
(Figures in Millions of Cable Passings)
Events
Has Cable
Does
Televisio
Not Have n
Cable
Service
Television Service , A
Total 9.8
Has cable
Internet Service , B
6.5
3.3
Does Not Have
Internet Service , B
5.9
11.7
17.6
Total
12.4
15.0
27.4
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4-25