Ch5 Probability
Ch 5.1 Probability Rule
Objective A : Sample Spaces and Events
Experiment – any activity that leads to well-defined results called outcomes.
Outcome – the result of a single trial of a probability experiment.
Sample space, S – the set of all possible outcomes of a probability experiment.
Event, E – a subset of the sample space
Simple event, ei – an event with one outcome is called a simple event.
Compound event – consists of two or more outcomes.
Example 1: A die is tossed one time.
(a) List the elements of the sample space S.
(b) List the elements of the event consisting of a number that is greater than 4.
Example 2: A coin is tossed twice. List the elements of the sample space S, and list the
elements of the event consisting of at least one head.
Objective B : Requirements for Probabilities
1. Each probability must lie on between 0 and 1. ( 0  P( E )  1)
2. The sum of the probabilities for all simple events in S equals 1.
(  P(ei )  1)
If an event is impossible, the probability of the event is 0.
If an event is a certainty, the probability of the event is 1.
An unusual event is an event that has a low probability of occurring. Typically, an event with a
probability less than 0.05 is considered as unusual.
Probabilities should be expressed as reduced fractions or rounded to three decimal places.
Example 1: A probability experiment is conducted. Which of these can be considered a
probability of an outcome?
(a) 2/5
(b) -0.28
(c) 1.09
Example 2: Why is the following not a probability model?
Color
Red
Green
Yellow
Probability
0.28
0.56
0.37
Example 3: Given: 𝑆 = {𝑒1 , 𝑒2 , 𝑒3 , 𝑒4 }
𝑃(𝑒1 ) = 𝑃(𝑒2 ) = 0.2 and 𝑃(𝑒3 ) = 0.5
Find: 𝑃(𝑒4 )
Objective C : Calculating Probabilities
C1. Approximating Probabilities Using the Empirical Approach
(Relative Frequency Approximation of Probability)
The probability of event E is approximately the number of times event E is observed divided by the
number of repetitions of the experiment.
P( E )  Relative Frequency of E 
Frequency of E
Number of Trials of Experiement
Example 1: Suppose that you roll a die 100 times and get six 80 times. Based on these results, what is
the probability that the next roll results in six?
Example 2: During a sale at men’s store, 16 white sweaters, 3 red sweaters, 9 blue sweaters, and 7
yellow sweaters were purchased. If a customer is selected at random, find the probability
that he bought a sweater that was not white.
Example 3: The age distribution of employees for this college is shown below:
Age
Under 20
20 – 29
30 – 39
40 – 49
50 and over
Number of Employees
25
48
32
15
10
If an employee is selected at random, find the probability that he or she is in the following
age groups
(a) Between 30 and 39 years of age
(b) Under 20 or over 49 years of age
C2. Classical Approach to Probability
(Equally Likely Outcomes are required)
If an experiment has n equally likely outcomes and if the number of ways that an event E can occur in
m , then the probability of E , P ( E ) , is
P( E ) 
Number of ways that E can occur m

Number of possible outcomes
n
If S is the sample space of this experiment,
N (E)
P( E ) 
N (S )
where N ( E ) is the number of outcomes in event E , and N ( S ) is the number of outcomes in the sample
space.
Example 1: Let the sample space be S  1, 2,3, 4,5,6,7,8,9,10 .
Suppose the outcomes are equally likely.
(a) Compute the probability of the event F  5,9 .
(b) Compute the probability of the event E = "an odd number."
Example 2: Two dice are tossed. Find the probability that the sum of two dice is greater than 8?
Example 3: If one card is drawn from a deck, find the probability of getting
(a) a club; (b) a 4 and a club
Example 4: Three equally qualified runners, Mark, Bill, and Alan, run a 100-meter sprint, and the order
of finish is recorded.
(a) Give a sample space S .
(b) What is the probability that Mark will finish last?
Ch 5.2 The Addition Rules and Complements
Objective A : Addition Rule for Disjoint (Mutually Exclusive) Events
Event A and B are disjoint (mutually exclusive) if they have no outcomes in common.
Addition Rule for Disjoint Events
If E and F are disjoint events, then P( E or F )  P( E )  P( F ) .
Example 1. A standard deck of cards contains 52 cards. One card is randomly selected
from the deck. Compute the probability of randomly selecting a two or three from
a deck of cards.
Objective B : General Addition Rule
The General Addition Rule
For any two events E and F , P( E or F )  P ( E )  P ( F )  P ( E and F ) .
Example 1 : A standard deck of cards contains 52 cards. One card is randomly selected from
the deck. Compute the probability of randomly selecting a two or club from a deck
of cards.
Objective C : Complement Rule
Complement Rule
If E represents any event and E c represents the complement of E , then
P( E C )  1  P( E )
e.g. The chance of raining tomorrow is 70%. What is the probability that it will not rain
tomorrow?
Example 1: A probability experiment is conducted in which the sample space of the experiment
is S  1, 2,3, 4,5,6,7,8,9,10,11,12 .
Let event E  2,3,5,6,7 , event F  5,6,7,8 , and event G  9,11
(a) List the outcome in E and F . Are E and F mutually exclusive?
(b) Are F and G mutually exclusive? Explain.
(c) List the outcome in E or F . Find P( E or F ) by counting the number of outcomes
in E or F .
(d) Determine P( E or F ) using the General Addition Rule.
(e) List the outcomes in E C . Find P( E C ) by counting the number of outcomes in E C .
(f) Determine P( E C ) using the Complement Rule.
Example 2: In a large department store, there are 2 managers, 4 department heads, 16 clerks,
and 4 stock persons. If a person is selected at random,
(a) find the probability that the person is a clerk or a manager;
(b) find the probability that the person is not a clerk.
Example 3: The following probability shows the distribution for the number of rooms in U.S.
housing units.
(a) Verify that this is a probability model.
(b) What is the probability that a randomly selected housing unit has four or more
rooms? Interpret this probability.
Example 4: According to the U.S. Census Bureau, the probability that a randomly selected
household speaks only English at home is 0.81. The probability that a randomly
selected household speaks only Spanish at home is 0.12.
(a) What is the probability that a randomly selected household speaks only English or
only Spanish at home?
(b) What is the probability that a randomly selected household speaks a language
other than only English at home?
(c) Can the probability that a randomly selected household speaks only Polish at home
equal 0.08? Why or why not?
Objective D : Contingency Table
A contingency table relates two categories of data. It is also called a two-way table which
consists of a row variable and a column variable. Each box inside the table is called a cell.
Example 1: In a certain geographic region, newspapers are classified as being published daily
morning, daily evening, and weekly. Some have a comics section and other do not.
The distribution is shown here.
Have comics
Section
Morning
Evening
Weekly
Yes
2
3
1
No
3
4
2
If a newspaper is selected at random, find these probabilities.
(a) The newspaper is a weekly publication.
(b) The newspaper is a daily morning publication or has comics.
(c) The newspaper is published weekly or does not have comics.
Ch 5.3 Independence and the Multiplication Rule
Objective A : Independent Events
Two events are independent if the occurrence of event E does not affect the probability of
event F .
Two events are dependent if the occurrence of event E affects the probability of event F .
Example 1: Determine whether the events E and F are independent or dependent. Justify
your answer.
(a) E : The battery in your cell phone is dead.
F : The battery in your calculator is dead.
(b) E : You are late to class.
F : Your car runs out of gas.
Objective B : Multiplication Rule for Independent Events
If E and F are independent events, then P( E and F )  P( E )  P( F )
Example 1: If 36% of college students are underweight, find the probability that if three college
students are selected at random, all will be underweight.
Example 2: If 25% of U.S. federal prison inmates are not U.S. citizens, find the probability that
two randomly selected federal prison inmates will be U.S. citizens.
Objective C : At-Least Probabilities
Probabilities involving the phrase “at least” typically use the Complement Rule. The phrase at
least means “greater than or equal to.” For example, a person must be at least 17 years old to
see an R-rated movie.
Example 1: If you make random guesses for four multiple-choice test questions (each with five
possible answers), what is the probability of getting at least one correct?
Example 2: For the fiscal year 2007, the IRS audited 1.77% of individual tax returns with
income of $100,000 or more. Suppose this percentage stays the same for the
current fiscal year.
(a) Would it be unusual for a return with income of $100,000 or more to be audited?
(b) What is the probability that two randomly selected returns with income of
$100,000 or more will be audited?
(c) What is the probability that two randomly selected returns with income of
$100,000 or more will NOT be audited?
(d) What is the probability that at least one of the two randomly selected returns with
income $100,000 or more will be audited?