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CHAPTER 5
Discrete Probability Distribution
Objectives
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Construct a probability distribution for a random variable.
Find the mean, variance, and expected value for a discrete random variable.
Find the exact probability for X successes in n trials of a binomial experiment.
Find the mean, variance, and standard deviation for the variable of a binomial
distribution.
Find probabilities for outcomes of variables using the Poisson, hypergeometric, and
multinomial distributions.
5. 1 Introduction
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Many decisions in business, insurance, and other real-life situations are made by
assigning probabilities to all possible outcomes pertaining to the situation and then
evaluating the results.
This chapter explains the concepts and applications of probability distributions. In
addition, special probability distributions, such as the binomial, multinomial, Poisson,
and hypergeometric distributions are explained.
5.2 Probability Distribution
I. Random Variables
A random variable is a variable whose values are determined by chance.
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A random variable is discrete if it can potentially assume only a finite or countable
number of values.
A random variable is continuous if it potentially can take on any value on an interval.
Example 1: Identify the following random variables as discrete or continuous.
a)
Weight of a package
b)
Number of students in a first-grade classroom
c)
Age of a cancer patient
Example 2:
An experiment consists of tossing five fair coins. Let x be the random variable that is
the number of heads in the five tosses. Is x discrete or continuous? List the sample
space of values for x.
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Example 3:
Let x be the random variable that gives the amount of time it takes a person to drive
to work. Is x discrete or continuous? List the sample space of values of x.
II. Discrete Probability Distribution
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A discrete probability distribution consists of the values a random variable can
assume and the corresponding probabilities of the values. The probabilities are
determined theoretically or by observation.
Example 1:
The probabilities that a customer will purchase 0, 1, 2, or 3 books are 0.45, 0.30,
0.15, and 0.10, respectively.
(a) Construct a probability distribution for the data.
(b) Draw a graph for the distribution.
Example 2:
Consider families with three children. Let x be the number of girls in a family.
Find the probability distribution for x and construct a graph for the probability
distribution.
III. Requirements for a Discrete Probability Distribution
1. P(x) will always be a number between 0 and 1 inclusive:
0  P(x)  1
2. The sum of the values of P(x) for each distinct value of x is 1:

P(x) = 1
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Example 1:
A random variable x has this probability distribution:
x
P(x)
0
.2
1
.3
2
.1
3
?
(a) Find P(x = 3)
(b) What is the probability that x is greater than 0?
(c) What value of x is most likely to occur?
(d) What is P(x = 8) ?
Example 2:
Determine whether the distribution represents a probability distribution.
If it does not, state why.
x
P(x)
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1.2
10
0.3
15
0.5
5.3 Mean, Variance, Standard Deviation, and Expectation
I. The mean of a discrete probability distribution
In order to find the mean for a probability distribution, one must multiply each
possible outcome by its corresponding probability and find the sum of the products.
   x  P( x)
where X1, X2, X3, …,Xn are the outcomes and
P(X1), P(X2), P(X3),…P(Xn) are the corresponding probability.
II. Variance of a Probability Distribution
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The variance of a probability distribution is found by multiplying the square of each
outcome by its corresponding probability, summing those products, and subtracting
the square of the mean.
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(x  )
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The formula for calculating the variance is:  2 
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The formula for the standard deviation is:    2
2
 P( x)
Example 1:
For the probability distribution given below, find (a) the mean, (b) the variance,
and (c) the standard deviation. Also (d) construct a graph for the probability
distribution and describe the shape of the distribution.
x
P(x)
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a)
b)
c)
d)
0
1/10
1
4/10
2
3/10
3
2/10
The mean
Variance
Standard deviation
Construct a graph for the probability distribution and describe the shape of
the distribution.
Example 2: Use the computational formula to find the standard deviation of the
probability distribution given in example 1.
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III. Expected Value
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Expected value or expectation is used in various types of games of chance, in
insurance, and in other areas, such as decision theory.
The expected value of a discrete random variable of a probability distribution is the
theoretical average of the variable. The formula is: E( x)   
x  P( x)

For a discrete random variable x, the expected value of x is the mean of the random
variable x.
The symbol E(X) is used for the expected value.
Example 1: (Ref: General Statistics by Chase/Bown, 4th Ed.)
A high school class decides to raise some money by conducting a raffle. The
students plan to sell 2000 tickets at $1 apiece. They will give one prize of
$100, two prizes of $50, and three prizes of $25. If you plan to purchase one
ticket, what are your expected net winnings?
Example 2: (Ref: Elementary Statistics by Triola, 9th edition)
In New Jersey’s Pick 4 lottery game, you pay 50 cents to select a sequence
of four digits, such as 7273. If you win by selecting the same sequence of
four digits that are drawn, you collect $2,788.
(a) How many different selections are possible?
(b) What is the probability of winning?
(c) If you win, what is your net profit?
(d) Find the expected value.
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5.4 The Binomial Distribution
I. Binomial Experiment
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Many types of probability problems have only two possible outcomes or they can be
reduced to two outcomes.
Examples include: when a coin is tossed it can land on heads or tails, when a baby
is born it is either a boy or girl, etc
The Binomial Experiment
The binomial experiment is a probability experiment that satisfies these
requirements:
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Each trial can have only two possible outcomes—success or
failure.
2.
There must be a fixed number of trials.
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The outcomes of each trial must be independent of each other.
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The probability of success must remain the same for each trial.
The outcomes of a binomial experiment and the corresponding probabilities of
these outcomes are called a binomial distribution
Example 1: (Ref: Exploring Statistics by Kitchens, 2nd ed.)
Which of the following are binomial random variables?
(a) The number of successful heart transplants out of five patients.
(b) The length of a prison term for possession of marijuana
(c) The name of each student in Math 227
(d) The number of approved food stamp recipients out of 50 applications
II. Notation for the Binomial Distribution
• The symbol for the probability of success
• The symbol for the probability of failure
• The numerical probability of success
• The numerical probability of failure
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The number of trials
The number of successes
III. Binomial Probability Formula
A binomial experiment consists of n identical trials with probability of success p on each
trial. The probability of x successes in n trials is
P( x) 
n
 p x  q n x
(n  x)! x !
for x  0,1, 2,....., n
OR
P( x)  n Cx  p x  q n  x
for x  0,1, 2,....., n
Example 1:
(a) Find
8C3
(b) Find
12C7
Example 2:
Consider a binomial experiment with n = 15, p = .3, and x = number of
success. Use the Binomial Formula for P(x) to find the probability that
(a) x = 11
(b) x is less than 2
Example 3:
It was found that 68% of American victims of health care fraud are senior
citizens. If 10 victims are randomly selected, find the probability that exactly
3 are senior citizens.
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Example 4: Consider a binomial experiment with n = 12, p = .4, and x = number of
success. Use the Bi Table to find the probability that
(a) x is greater than 5 but less than 8
(b) x is greater than 7
(c) x equals 6
IV. Mean and Variance of a Binomial Distribution
For a binomial experiment consisting of n trials with the probability of success p, the
mean or expected of x is
For a binomial experiment consisting of n trials with the probability of success p, the
variance of x is
The standard deviation of x is
Example 1:
Assume that 60% of a college’s student loan applications are approved.
Ten applications are chosen at random.
(a) What is the probability that eight or more are approved?
(b)
How many applications are expected to be approved?
(Find the mean of the number approved out of ten applications.)
(c) What is the standard deviation of the number approved out of ten
applications?
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Summary
• A probability distribution can be graphed, and the mean, variance, and standard
deviation can be found.
• The mathematical expectation can also be calculated for a probability distribution.
• Expectation is used in insurance and games of chance.
• The binomial distribution is used when there are only two outcomes for an
experiment, a fixed number of trials, the probability is the same for each trial, and the
outcomes are independent of each other.
Conclusion
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Many decisions in business, insurance, and other real-life situations are made
by assigning probabilities to all possible outcomes pertaining to the situation
and then evaluating the results.
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