CK-12 Probability and Statistics - Advanced

4.4. The Binomial Probability Distribution http://www.ck12.org

over a million possible outcomes. To tabulate such outcomes would be impractical. Fortunately, there is a formula
for the binomial distribution that saves us all the numerous calculations.

If annexperiments are performed, then the number of ways to get exactlyxsuccessesSis equal to the binomial
coefficients. In other words the event of obtaining exactlyxsuccessesSin thentrials consists ofCxnoutcomes. From
previous chapters, recall that

Cnx=

(

n

x

)

=

n!

x!(n−x)!

is the number of simple events that havexsuccesses and(n−x)failures. Here in this chapter, we introduce the
above notation

(n

x

)

that is equivalent toCxnof Chapter 3. Both notations are used interchangeably in statistics and it
is a good idea to be familiar with both.

The Binomial Probability Distribution

Supposenexperiments are performed, with the probability of successes on any given trial isp. Letxdenote the
total number of successes in thentrials. Then the probability distribution of the random variablexis given by the
formula,

p(x) =

(

n

x

)

px( 1 −p)n−x=

(

n

x

)

pxqn−x

To apply the binomial formula to a specific problem, it is useful to have an organized strategy. Such a strategy is
presented in the following steps:

1. Identify a success.


2. Determinep, the success probability.


3. Determinen, the number of experiments or trials.


4. Use the binomial formula to write the probability distribution ofx.

The examples below will help you learn how to use the binomial formula.

Example:

According to a study conducted by a telephone company, the probability is 25% that a randomly selected phone call
will last longer than the mean value of 3.8 minutes. What is the probability that out of three randomly selected calls

a. exactly two last longer than 3.8 minutes?

b. None last longer than 3.8 minutes?

Solution:

Showing the four steps listed above.

1. The success is any call that is longer than 3.8 minutes.


2. The probabilityp=25%= 0 .25.


3. The number of trialsnis 3.


4. Thus we can now use the binomial probability formula,