CHAPTER 4: DISCRETE PROBABILITY DISTRIBUTIONS
4.1 Probability Distribution
random variable: a random variable x represents a numerical value associated with
each outcome of a probability experiment
discrete: a random variable is discrete if it has a finite or countable number of possible
outcomes that can be listed
continuous: a random variable is continuous if it has an uncountable number of possible
outcomes, represented by an interval on the number line
Example 1: Decide whether the random variable x is discrete or continuous:
1. x represents the length of time it takes to complete a test
2. x represents the number of songs played by a band at a rock festival
discrete probability distribution: lists each possible value the random variable can
assume, together with its probability, a probability distribution must satisfy the
following conditions:
1. The probability of each value of the discrete random variable is between 0 and
1, inclusive. 0 < P(x) < 1
2. The sum of all the probabilities is 1. P(x) = 1
Constructing a Discrete Probability Distribution:
Let x be a discrete random variable with possible outcomes x1, x2,…. xn
1. Make a frequency distribution for the possible outcomes.
2. Find the sum of the frequencies.
3. Find the probability of each possible outcome by dividing its frequency by the sum of
the frequencies.
4. Check that each probability is between 0 and 1, inclusive, and that the sum is 1.
Example 2: A company tracks the number of sales new employees make each day during a 100day probationary period. The results for one new employee are shown below. Construct and
graph a probability distribution.
Sales per day, x
Number of days, f
0
16
1
19
2
15
3
21
4
9
5
10
6
8
7
2
Probability
Example 3: Verify that the distribution constructed in Example 2 is a probability distribution.
Example 4: Decide whether the distribution is a probability distribution. Explain your
reasoning.
x
5
6
7
8
x
1
2
3
4
P(x)
1/16
5/8
1/4
1/8
P(x)
0.09
0.36
0.49
0.06
Mean of a Discrete Random Variable: the mean of a discrete random variable is given by
   xP(x) . Each value of x is multiplied by its corresponding probability and the products
are added.
Example 5: Find the mean of the probability distribution constructed in Example 2. What can
you conclude?
x
P(x)
0
.16
1
.19
2
.15
3
.21
4
.09
5
.10
6
.08
7
.02
xP(x)
Standard Deviation of a Discrete Random Variable: the variance of a discrete
random variable is  2   ( x   ) 2 P( x)
.
The standard deviation is    2 
 (x  )
2
P( x)
Example 6: Find the variance and standard deviation for the probability distribution constructed
in Example 2.
x
P(x)
0
.16
1
.19
2
.15
3
.21
4
.09
5
.10
6
.08
7
.02
Expected value: the expected value of a discrete random variable is equal to the mean of the
random variable. Expected value = E(x) = E( x)     xP( x)
Example 7: At a raffle, 2000 tickets are sold at $5 each for five prizes of $2000, $1000, $500,
$250, and $100. You buy one ticket. What is the expected value of your gain? Use a
probability distribution to calculate the expected value.
4.2 Binomial Distributions
Binomial experiment: A binomial experiment is a probability experiment that satisfies the
following conditions:
1. The experiment is repeated for a fixed number of trials, where each trial is independent of
the other trials.
2. There are only two possible outcomes of interest for each trial. The outcomes can be
classified as a success (S) or as a failure (F).
3. The probability of a success P(S) is the same for each trial.
4. The random variable x counts the number of successful trials.
Notation for Binomial Experiments:
n
The number of times a trial is repeated.
p = P(S)
The probability of success in a single trial.
q = P(F)
The probability of failure in a single trial (q = 1 – p).
x
The random variable represents a count of the number of successes in n
trials: x = 0, 1, 2, 3, …, n.
Example 1: Decide whether the following is a binomial experiment. If it is, specify the values
of n, p, and q, and list the possible values of the random variable x. If it is not, explain why.
You take a multiple-choice quiz that consists of 10 questions. Each question has four possible
answers, one of which is correct. To complete the quiz, you randomly guess the answer to each
question. The random variable represents the number of correct answers.
Binomial Probability Formula: In a binomial experiment, the probability of exactly x
n!
successes in n trials is P( x)  n C x p x q n  x 
p x q n x
(n  x)! x!
Example 2: A card is selected from a standard deck and replaced. This experiment is repeated a
total of five times. Find the probability of selecting exactly three clubs.
binomial probability distribution: you can construct a binomial probability distribution
by listing the possible values of x with the corresponding probability of each
Example 3: In a survey, workers in the United States were asked to name their expected source s
of retirement income. (The results are shown in the graph on page 209.) Seven workers who
participated in the survey are randomly selected and asked whether they expect to rely on a
pension for retirement income. Create a binomial probability distribution for the number of
retirees who respond yes.
Example 4: The results of a recent survey indicate that 71% of people in the United States use
more than one topping on their hot dogs. If you randomly select 250 people, what is the
probability that exactly 178 of them will use more than one topping? Use your calculator to find
the probability.
[2nd, vars, binomialpdf(n,p,x)]
Example 5: A survey indicates that 21% of men in the United States consider fishing their
favorite leisure-time activity. You randomly select five U.S. men and ask them if fishing is their
favorite leisure-time activity. Find the probability that
1. Exactly two of them respond yes
2. At least two of them respond yes
3. Fewer than two of them respond yes
Example 6: Forty-five percent of all small businesses in the United States have a web site. If
you randomly select 10 small businesses, what is the probability that exactly four of them have a
web site? Use a binomial distribution table to find the probability.
Example 7: Sixty-two percent of households in the United States own a computer. You
randomly select six households and ask if they own a computer. Construct a probability
distribution for the random variable x. Then graph the distribution.
Population Parameters of a Binomial Distribution:
Mean:   np
Variance:  2  npq
Standard Deviation:   npq
Example 8: In San Francisco, California, 44% of the days in a year are clear. Find the mean,
variance, and standard deviation for the number of clear days during the month of May. Interpret
the results and determine any unusual values.
4.3 More Discrete Probability Distributions
Geometric Distribution: a discrete probability distribution of a random variable x that
satisfies the following conditions.
1. A trial is repeated until a success occurs.
2. The repeated trials are independent of each other.
3. The probability of success p is constant for each trial.
The probability that the first success will occur on trial number x is
P( x)  p(q) x 1 , where q = 1 – p.
Example 1: From experience, you know that the probability that you will make a sale on any
given telephone call is 0.23. Find the probability that your first sale will occur before your
fourth sales call.
Poisson Distribution: a discrete probability distribution of a random variable x that
satisfies the following conditions.
1. The experiment consists of counting the number of times, x, an event occurs in a
given interval. The interval can be an interval of time, area, or volume.
2. The probability of the event occurring is the same for each interval.
3. The number of occurrences in one interval is independent of the number of
occurrences in other intervals.
The probability of exactly x occurrences in an interval is P( x) 
 x e 
, where e is an
x!
irrational number approximately equal to 2.71828 and  is the mean number of
occurrences per interval unit.
Example 2: The mean number of accidents per month at a certain intersection is three. What is
the probability that in any given month four accidents will occur at this intersection? What is the
probability that more than four accidents will occur in any given month at the intersection?
Example 3: Two thousand brown trout are introduced into a small lake. The lake has a volume
of 20,000 cubic meters. Use a table to find the probability that three brown trout are found in an
given cubic meter of the lake.
**There is a table with a summary of discrete probability distributions on page 225!!