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PQ220 6234F.Ch 03 13/04/2002 03:19 PM Page 74

74 CHAPTER 3 DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

To complete a general probability formula, only an expression for the number of outcomes
that contain x errors is needed. An outcome that contains x errors can be constructed by parti-
tioning the four trials (letters) in the outcome into two groups. One group is of size x and
contains the errors, and the other group is of size n x and consists of the trials that are okay.
The number of ways of partitioning four objects into two groups, one of which is of size x, is
4 4!
a b . Therefore, in this example
x x!14 x2!

4
x
P1X x2 a b 10.12 10.92 4 x
x
4
Notice that a b 4! 32! 2!4 6 , as found above. The probability mass function of X
2
was shown in Example 3-4 and Fig. 3-1.

The previous example motivates the following result.

Definition
A random experiment consists of n Bernoulli trials such that
(1) The trials are independent
(2) Each trial results in only two possible outcomes, labeled as “success’’ and
“failure’’
(3) The probability of a success in each trial, denoted as p, remains constant

The random variable X that equals the number of trials that result in a success
has a binomial random variable with parameters 0 p 1 and n 1, 2, p. The
probability mass function of X is

n x n x
f 1x2 a b p 11 p2 x 0, 1, p , n (3-7)
x

n
As in Example 3-16, a b equals the total number of different sequences of trials that
x
contain x successes and n x failures. The total number of different sequences that contain x
successes and n x failures times the probability of each sequence equals P1X x2.
The probability expression above is a very useful formula that can be applied in a num-
ber of examples. The name of the distribution is obtained from the binomial expansion. For
constants a and b, the binomial expansion is

n n
n
k n k
1a b2 a a b a b
k 0 k
Let p denote the probability of success on a single trial. Then, by using the binomial
expansion with a p and b 1 p, we see that the sum of the probabilities for a bino-
mial random variable is 1. Furthermore, because each trial in the experiment is classified
into two outcomes, {success, failure}, the distribution is called a “bi’’-nomial. A more