Introductory Biostatistics

objects,

n

x



¼

n!

x!ðn
xÞ!

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

x
np

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