Download Multiplication Rules

CHAPTER 4
Probability and Counting Rules
Objectives
• Determine sample spaces and find the probability of an event using classical
probability or empirical probability.
• Find the probability of compound events using the addition rules.
• Find the probability of compound events using the multiplication rules.
• Find the conditional probability of an event.
• Determine the number of outcomes of a sequence of events using a tree diagram.
• Find the total number of outcomes in a sequence of events using the fundamental
counting rule.
• Find the number of ways r objects can be selected from n objects using the
permutation rule.
• Find the number of ways r objects can be selected from n objects without regard to
order using the combination rule.
• Find the probability of an event using the counting rules.
Section 4.1 Introduction
• Probability as a general concept can be defined as the chance of an event occurring.
In addition to being used in games of chance, probability is used in the fields of
insurance, investments, and weather forecasting, and in various areas.
• Rules such as the fundamental counting rule, combination rule and permutation rules
allow us to count the number of ways in which events can occur.
• Counting rules and probability rules can be used together to solve a wide variety of
problems.
Section 4.2 Sample Spaces and Probability
I. Basic Concepts
• A probability experiment is a chance process that leads to well-defined results called
outcomes.
• An outcome is the result of a single trial of a probability experiment.
• A sample space is the set of all possible outcomes of a probability experiment.
Sample Spaces and Events
Examples of some sample space:
Experiment
Toss one coin
Roll a die
Answer a true/false question
Toss two coins
Sample Space
An event consists of a set of outcomes of a probability experiment.
Simple event - an event with one outcome
Compound event - an event with two or more outcomes.
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Basic Concepts (cont.)
•
•
Equally likely events are events that have the same probability of occurring.
A tree diagram is a device consisting of line segments emanating from a starting point
and also from the outcome point. It is used to determine all possible outcomes of a
probability experiment.
Example: A tree diagram to find the sample space for tossing two coins.
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.
Example 3: A randomly selected citizen is interviewed and the following information is
recorded: employment status and level of education.
The symbols for employment status are
Y = employed and N = unemployed, and the symbols for level of education are
1 = did not complete high school, 2 = completed high school but did not complete
college, and 3 = completed college. List the elements of the sample space, and
list the elements of the following events:
(a) Did not complete high school
(b) Is unemployed
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II.
Calculating Probabilities
A. Empirical Probability
(Relative Frequency Approximation of Probability)
•
Empirical probability relies on actual experience to determine the likelihood of
outcomes.
•
Given a frequency distribution, the probability of an event being in a given class
is: 
Example 1: The age distribution of employees for this college is shown below:
Age
Under 20
20 – 29
30 – 39
40 – 49
50 and over
# 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
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.
B. Classical Probability
• Classical probability uses sample spaces to determine the numerical probability that
an event will happen.
• Classical probability assumes that all outcomes in the sample space are equally likely
to occur.

Example 1:
A statistics class contains 14 males and 20 females.
A student is to be selected by chance and the gender of the student
recorded.
(a) Give a sample space S for the experiment.
(b) Is each outcome equally likely? Explain.
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(c) Assign probabilities to each outcome? (Ref: General Statistics by
Chase/Bown, 4th Ed.)
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 100meter sprint, and the order of finish is recorded.
(a) Give a sample space S .
(b) What is the probability that Mark will finish last?
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III. Rounding Rule for Probabilities
•
Probabilities should be expressed as reduced fractions or rounded to two or three
decimal places. W hen the probability of an event is an extremely small decimal, it is
permissible to round the decimal to the first nonzero digit after the decimal point.
Probability Rules
1. The probability of an event E is a number (either a fraction or decimal) between and
including 0 and 1. This is denoted by: 
*Rule 1 states that probabilities cannot be negative or greater than one.
2. If an event E cannot occur (i.e., the event contains no members in the sample space),
the probability is zero.
3. If an event E is certain, then the probability of E is 1.
4. The sum of the probabilities of the outcomes in the sample space is 1. 
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: Given: S  {E1 , E2 , E3 , E4 }
P( E1 )  P( E2 )  0.2 and P( E3 )  0.5
Find: P( E4 )
IV. Complementary Events
•
•
The complement of an event E is the set of outcomes in the sample space that are
not included in the outcomes of event E. The complement of E is denoted by
.
Rule for Complementary Events

OR
Example 1 : The chance of raining tomorrow is 70%. What is the probability that it will
not rain tomorrow?
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Section 4.3 The Addition Rules for Probability
Mutually Exclusive Events
•
Two events are mutually exclusive if they cannot occur at the same time (i.e.,
they have no outcomes in common).
•
The probability of two or more events can be determined by the addition rules.
I. Addition Rules
•
Addition Rule 1—W hen two events A and B are mutually exclusive, the probability
that A or B will occur is: 
Ex1) A single card is drawn from a deck. Find the probability of selecting a club or a
diamond.
P (club or diamond)  mutually exclusive
P (club or diamond) = P (club) + P (diamond)
Ex2) 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, find the probability that the
person is either a clerk or a manager.
I. Addition Rules (cont.)
•
Addition Rule 2—If A and B are not mutually exclusive, then:

Example 1: A single card is drawn from a deck. Find the probability of selecting a jack or
a black card.
Example 2: 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.
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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.
Section 4.4 The Multiplication Rules and Conditional Probability
Independent Events
Two events A and B are independent if the fact that A occurs does not affect the
probability of B occurring.
When the outcome or occurrence of the first event affects the outcome or occurrence
of the second event in such a way that the probability is changed, the events are said
to be dependent.
I. Multiplication Rules
•
•
The multiplication rules can be used to find the probability of two or more events that
occur in sequence.
Multiplication Rule 1—When two events are independent, the probability of both
occurring is: 
Example 1: If 36% of college students are overweight, find the probability
that if three college students are selected at random, all will be
overweight.
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.
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Example 3: Suppose the probability of remaining with a particular company 10
years or longer is 1/6. A man and a woman start work at the
company on the same day.
(a) What is the probability that the man will work there less than 10
years?
(b) What is the probability that both the man and the woman will
work there less than 10 years?
Example 4: A smoke-detector system uses two devices, A and B. If smoke is
present, the probability that it will be detected by device A is .95;
by device B, .98; and by both devices, .94.
(a) If smoke is present, find the probability that the smoke will be
detected by device A or device B or both devices.
(b) Find the probability that the smoke will not be detected.
Example 5: 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?
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Example 6: There are 2000 voters in a town. Consider the experiment of
randomly selecting a voter to be interviewed.
The event A consists of being in favor of more stringent building
codes; the event B consists of having lived in the town less than
10 years. The following table gives the numbers of voters in
various categories. (Ref: General Statistics by Chase/Bown,
4th Ed.)
A
Favor more
stringent codes
B Less than 10 years
B At least 10 years
100
1000
A
Do not favor more
stringent codes
700
200
Find each of the following:
(a) P(A)
(b) P( B )
(c) P(A and B)
Multiplication Rules (cont.)
Multiplication Rule 2—When two events are dependent (e.g. the outcome of event A
affects the outcomes of event B), the probability of both occurring is:

Example 1:
A box has 5 red balls and 2 white balls. If two balls are randomly selected
(one after the other), what is the probability that they both are red?
(a) with replacement
(b) without replacement
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Example 2:
Three cards are drawn from a deck without replacement. Find the
probability that all are jacks.
Multiplication Rules (cont.)
•
The conditional probability of an event B in relationship to an event A is the probability
that event B occurs after event A has already occurred. The notation for conditional
probability is P(B|A).
Formula for Conditional Probability
• The probability that the second event B occurs given that the first event A has
occurred can be found dividing the probability that both events occurred by the
probability that the first event has occurred. The formula is: 
Example 1: Two fair dice are tossed. Consider the following events:
A = sum is 7 or more,
B = sum is even,
and C = a match (both numbers are the same).
(a) P(A and B)
(b) P(A or B)
(c) P(AB)
(d) P(BC)
Example 2: At a local Country Club, 65% of the members play bridge and swim,
and 72% play bridge. If a member is selected at random, find the
probability that the member swims, given that the member plays
bridge.
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Example 3: Eighty students in a school cafeteria were asked if they favored a
ban on smoking in the cafeteria. The results of the survey are
shown in the table.
Class
Favor
Oppose
No opinion
Freshman
15
27
8
Sophomore
23
5
2
If a student is selected at random, find these probabilities.
(a) The student is a freshman or favors the ban.
(b) Given that the student favors the ban, the student is a sophomore.
Section 4.5 Counting Rules
• The Fundamental Counting Rule
In a sequence of n events in which the first one has k1 possibilities and the second
event has k2 and the third has k3, and so forth, the total number of possibilities of the
sequence will be: 
Note: “And” in this case means to multiply.
Example 1: Two dices are tossed. How many outcomes are in S.
Example 2: A password consists of two letters followed by one digit. How
many different passwords can be created? (Note: Repetitions
are allowed)
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Example 3: Suppose four digits are to be randomly selected
(with repetitions allowed).
(Note: The set of digits is {0,1,2,3,4,5,6,7,8,9}.)
If the first digit must be a 2 and repetitions are permitted,
How many different 4-digit can be made?
II. Permutations and Combinations
•
Factorial Notation
5! = 5·4·3·2·1
7! = 7·6·5·4·3·2·1
In general,
n! = n · (n-1) ·(n-2) ·(n-3) ···· 1
0!=1
Permutation
• A permutation is an arrangement of n objects in a specific order.
The arrangement of n objects in a specific order using r objects at a time is called a
permutation of n objects taking r objects at a time.
Note: Order does matter.
It is written as nPr, and the formula is: 
Combinations
• A selection of distinct objects without regard to order is called a combination.
(Order does NOT matter!)
•
The number of combinations of r objects selected from n objects is denoted nCr and is
given by the formula: 
Note: Combinations are always less than permutations for the same n and r.
Example 1:
Find (a) 5!
(b)
25
P8
(c)
12
P4
(d)
C8
25
12
Example 2: A television news director wishes to use three news stories on
an evening show. One story will be the lead story, one will be
the second story, and the last will be a closing story. If the
director has a total of eight stories to choose from, how many
possible ways can the program be set up?
Example 3: If a person can select 3 presents from 10 presents, how many
different combinations are there?
Example 4: Your family vacation involves a cross-country air flight, a rental
car, and a hotel stay in Boston. If you can choose from four major
air carriers, five car rental agencies, and three major hotel chains,
how many options are available for your vacation
accommodations?
Section 4.6 Probability and Counting Rules
By using the fundamental counting rule, the permutation rules, and the combination rule,
the probability of outcomes of many experiments can be computed.
Example 1:
A combination lock consists of the 26 letters of the alphabet.
If a three-letter combination is needed, find the probability
that the combination will consist of the first two letters AB
in that order. The same letter can be used more than once.
Example 2:
Five cards are selected from a 52-card deck for a poker hand.
(a) How many outcomes are in the sample space?
(b) A royal flush is a hand that contains that A, K, Q, J, 10, all in
the same suit. How many ways are there to get a royal flush?
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(c) What is the probability of being dealt a royal flush?
Summary
•
•
•
•
•
•
•
•
•
•
•
The three types of probability are classical, empirical, and subjective.
Classical probability uses sample spaces.
Empirical probability uses frequency distributions and is based on observations.
In subjective probability, the researcher makes an educated guess about the chance
of an event occurring.
An event consists of one or more outcomes of a probability experiment.
Two events are said to be mutually exclusive if they cannot occur at the same time.
Events can also be classified as independent or dependent.
If events are independent, whether or not the first event occurs does not affect the
probability of the next event occurring.
If the probability of the second event occurring is changed by the occurrence of the
first event, then the events are dependent.
The complement of an event is the set of outcomes in the sample space that are not
included in the outcomes of the event itself.
Complementary events are mutually exclusive.
Summary
The number of ways a sequence of n events can occur; if the first event can occur in k1
ways, the second event can occur in k2 ways, etc.
(Multiplication rule)
The arrangement of n objects in a specific order using r objects at a time
(Permutation rule)
The number of combinations of r objects selected from n objects (order is not important)
(Combination rule)
Conclusions
•
•
Probability can be defined as the chance of an event occurring. It can be used to
quantify what the “odds” are that a specific event will occur. Some examples of how
probability is used everyday would be weather forecasting, “75% chance of snow” or
for setting insurance rates.
A tree diagram can be used when a list of all possible outcomes is necessary. W hen
only the total number of outcomes is needed, the multiplication rule, the permutation
rule, and the combination rule can be used.
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