objects,
n
x
¼
n!
x!ðnxÞ!
andn!is the product of the firstnintegers. For example,
3 !¼ð 1 Þð 2 Þð 3 Þ
The mean and variance of the binomial distribution are
m¼np
s^2 ¼npð 1 pÞ
and when the number of trialsnis from moderate to large (n>25, say), we we
approximate the binomial distribution by a normal distribution and answer
probability questions by first converting to a standard normal score:
z¼
xnp
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
npð 1 pÞ
p
wherepis the probability of having a positive outcome from a single trial. For
example, forp¼ 0 :1 andn¼30, we have
m¼ð 30 Þð 0 : 1 Þ
¼ 3
s^2 ¼ð 30 Þð 0 : 1 Þð 0 : 9 Þ
¼ 2 : 7
so that
Prðxb 7 ÞFPr zb
7 3
ffiffiffiffiffiffiffi
2 : 7
p
¼Prðzb 2 : 43 Þ
¼ 0 : 0075
In other words, if the true probability for having the side e¤ect is 10%, the
probability of having seven or more of 30 patients with the side e¤ect is less
than 1% (¼ 0 :0075).
PROBABILITY MODELS FOR DISCRETE DATA 135