A Survey of Probability Concepts

Chapter 5

Instructor:
Saad-Ur-Rehman
McGraw-Hill/Irwin ©The McGraw-Hill Companies, Inc. 2008
 GOALS

 Define probability.
 Describe the classical, empirical, and subjective
approaches to probability.
 Explain the terms experiment, event, outcome,
permutations, and combinations.
 Define the terms conditional probability and joint
probability.
 Calculate probabilities using the rules of addition
and rules of multiplication.
 Apply a tree diagram to organize and compute
probabilities.
 Calculate a probability using Bayes’ theorem.

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 Definitions

A probability is a measure of the likelihood

that an event in the future will happen. It
it can only assume a value between 0 and 1.
 A value near zero means the event is not
likely to happen. A value near one means it is
likely.
 There are three ways of assigning probability:
– classical,
– empirical, and
– subjective.
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 Probability Examples

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 Definitions continued

An experiment is the observation

of some activity or the act of taking
some measurement.
An outcome is the particular result
of an experiment.
An event is the collection of one or
more outcomes of an experiment.
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 Experiments, Events and Outcomes

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 Assigning Probabilities

Three approaches to assigning probabilities

– Classical
– Empirical
– Subjective

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 Classical Probability

Consider an experiment of rolling a six-sided die. What is the

probability of the event “an even number of spots appear face
up”?
The possible outcomes are:

There are three “favorable” outcomes (a two, a four, and a six) in

the collection of six equally likely possible outcomes.

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 Mutually Exclusive Events

 Events are mutually exclusive if the occurrence of any one event

means that none of the others can occur at the same time.
 Events are independent if the occurrence of one event does not
affect the occurrence of another.

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 Collectively Exhaustive Events

Events are collectively exhaustive

if at least one of the events must
occur when an experiment is
conducted.

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 Empirical Probability

The empirical approach to probability is based on what

is called the law of large numbers. The key to
establishing probabilities empirically is that more
observations will provide a more accurate estimate of
the probability.

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 Law of Large Numbers

Suppose we toss a fair coin. The result of each toss is either a

head or a tail. If we toss the coin a great number of times, the
probability of the outcome of heads will approach .5. The
following table reports the results of an experiment of flipping a
fair coin 1, 10, 50, 100, 500, 1,000 and 10,000 times and then
computing the relative frequency of heads

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 Empirical Probability - Example

On February 1, 2003, the Space Shuttle Columbia

exploded. This was the second disaster in 113 space
missions for NASA. On the basis of this information,
what is the probability that a future mission is
successfully completed?

Number of successful flights

Probability of a successful flight 
Total number of flights
111
 0.98
113

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 Subjective Probability - Example

 If there is little or no past experience or information on which to base a

probability, it may be arrived at subjectively.

 Illustrations of subjective probability are:

1. Estimating the likelihood the New England Patriots will play in the Super Bowl next
year.
2. Estimating the likelihood you will be married before the age of 30.
3. Estimating the likelihood the U.S. budget deficit will be reduced by half in the next 10
years.

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 Summary of Types of Probability

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 Rules for Computing Probabilities

Rules of Addition
 Special Rule of Addition - If two events
A and B are mutually exclusive, the
probability of one or the other event’s
occurring equals the sum of their
probabilities.
P(A or B) = P(A) + P(B)

 The General Rule of Addition - If A and

B are two events that are not mutually
exclusive, then P(A or B) is given by the
following formula:
P(A or B) = P(A) + P(B) - P(A and B)

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 Addition Rule - Example

What is the probability that a card chosen at

random from a standard deck of cards will be
either a king or a heart?

P(A or B) = P(A) + P(B) - P(A and B)

= 4/52 + 13/52 - 1/52
= 16/52, or .3077
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 The Complement Rule

The complement rule is used to determine

the probability of an event occurring by
subtracting the probability of the event not
occurring from 1.
P(A) + P(~A) = 1
or P(A) = 1 - P(~A).

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 Joint Probability – Venn Diagram

JOINT PROBABILITY A probability that

measures the likelihood two or more events
will happen concurrently.

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 Special Rule of Multiplication

 The special rule of multiplication requires that two events A and B are
independent.
 Two events A and B are independent if the occurrence of one has no effect on
the probability of the occurrence of the other.
 This rule is written: P(A and B) = P(A)P(B)

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 Multiplication Rule-Example

A survey by the American Automobile association

(AAA) revealed 60 percent of its members made
airline reservations last year. Two members are
selected at random. What is the probability both
made airline reservations last year?

Solution:
The probability the first member made an airline reservation last year
is .60, written as P(R1) = .60
The probability that the second member selected made a reservation is
also .60, so P(R2) = .60.
Since the number of AAA members is very large, you may assume that
R1 and R2 are independent.

P(R1 and R2) = P(R1)P(R2) = (.60)(.60) = .36

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 Conditional Probability

A conditional probability is the

probability of a particular event
occurring, given that another event
has occurred.
The probability of the event A given
that the event B has occurred is
written P(A|B).

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 General Multiplication Rule

The general rule of multiplication is used to find the joint probability that two events will occur.
Use the general rule of multiplication to find the joint probability of two events when the
events are not independent.
It states that for two events, A and B, the joint probability that both events will happen is
found by multiplying the probability that event A will happen by the conditional probability
of event B occurring given that A has occurred.

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 General Multiplication Rule - Example

A golfer has 12 golf shirts in his closet. Suppose 9 of these shirts are white
and the others blue. He gets dressed in the dark, so he just grabs a shirt
and puts it on. He plays golf two days in a row and does not do laundry.
What is the likelihood both shirts selected are white?

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 General Multiplication Rule - Example

 The event that the first shirt selected is white is W1.

The probability is P(W1) = 9/12
 The event that the second shirt selected is also white is
identified as W2. The conditional probability that the
second shirt selected is white, given that the first shirt
selected is also white, is P(W2 | W1) = 8/11.
 To determine the probability of 2 white shirts being
selected we use formula: P(AB) = P(A) P(B|A)
 P(W1 and W2) = P(W1)P(W2 |W1) = (9/12)(8/11) = 0.55

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 Contingency Tables

A CONTINGENCY TABLE is a table used to classify sample

observations according to two or more identifiable
characteristics

E.g. A survey of 150 adults classified each as to gender and the

number of movies attended last month. Each respondent is
classified according to two criteria—the number of movies
attended and gender.

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 Contingency Tables - Example

A sample of executives were surveyed about their loyalty to their company.

One of the questions was, “If you were given an offer by another
company equal to or slightly better than your present position, would
you remain with the company or take the other position?” The
responses of the 200 executives in the survey were cross-classified
with their length of service with the company.

What is the probability of randomly selecting an executive who is loyal to

the company (would remain) and who has more than 10 years of
service?
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 Contingency Tables - Example

Event A1 happens if a randomly selected executive will remain with

the company despite an equal or slightly better offer from
another company. Since there are 120 executives out of the
200 in the survey who would remain with the company
P(A1) = 120/200, or .60.
Event B4 happens if a randomly selected executive has more than
10 years of service with the company. Thus, P(B4| A1) is the
conditional probability that an executive with more than 10
years of service would remain with the company. Of the 120
executives who would remain 75 have more than 10 years of
service, so P(B4| A1) = 75/120.

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 Tree Diagrams

A tree diagram is useful for portraying

conditional and joint probabilities. It is
particularly useful for analyzing business
decisions involving several stages.
A tree diagram is a graph that is helpful in
organizing calculations that involve several
stages. Each segment in the tree is one stage of
the problem. The branches of a tree diagram are
weighted by probabilities.
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30
 Bayes’ Theorem

 Bayes’ Theorem is a method for

revising a probability given additional
information.
 It is computed using the following
formula:

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 Bayes Theorem - Example

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 Bayes Theorem – Example (cont.)

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 Bayes Theorem – Example (cont.)

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 Bayes Theorem – Example (cont.)

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 Bayes Theorem – Example (cont.)

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 Counting Rules – Multiplication

The multiplication formula indicates that if there are m ways of doing one thing and
n ways of doing another thing, there are m x n ways of doing both.
Example: Dr. Delong has 10 shirts and 8 ties. How many shirt and tie outfits does he have?
(10)(8) = 80

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 Counting Rules – Multiplication:
Example

An automobile dealer
wants to advertise that
for $29,999 you can buy
a convertible, a two-door
sedan, or a four-door
model with your choice
of either wire wheel
covers or solid wheel
covers. How many
different arrangements of
models and wheel
covers can the dealer
offer?

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 Counting Rules – Multiplication:
Example

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 Counting Rules - Permutation

A permutation is any arrangement of r

objects selected from n possible
objects. The order of arrangement is
important in permutations.

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 Counting - Combination

A combination is the number of

ways to choose r objects from a
group of n objects without regard
to order.

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 Combination - Example

There are 12 players on the Carolina Forest

High School basketball team. Coach
Thompson must pick five players among the
twelve on the team to comprise the starting
lineup. How many different groups are
possible?

12!
12 C5  792
5!(12  5)!
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 Permutation - Example

Suppose that in addition to selecting the group,

he must also rank each of the players in that
starting lineup according to their ability.

12!
12 P 5  95,040
(12  5)!

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 End of Chapter 5

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