Introductory Biostatistics

3.4.1 Binomial Distribution

In Chapter 1 we discussed cases with dichotomous outcomes such as male–
female, survived–not survived, infected–not infected, white–nonwhite, or
simply positive–negative. We have seen that such data can be summarized
into proportions, rates, and ratios. In this section we are concerned with the
probability of a compound event: the occurrence of x (positive) outcomes
ð 0 axanÞinntrials, called abinomial probability. For example, if a certain
drug is known to cause a side e¤ect 10% of the time and if five patients are
given this drug, what is the probability that four or more experience the side
e¤ect?
LetSdenote a side-e¤ect outcome andNan outcome without side e¤ects.
The process of determining the chance ofxS’s inntrials consists of listing all
the possible mutually exclusive outcomes, calculating the probability of each
outcome using the multiplication rule (where the trials are assumed to be inde-
pendent), and then combining the probability of all those outcomes that are
compatible with the desired results using the addition rule. With five patients
there are 32 mutually exclusive outcomes, as shown in Table 3.11.
Since the results for the five patients are independent, the multiplication rule
produces the probabilities shown for each combined outcome. For example:

The probability of obtaining an outcome with fourS’s and oneNis

ð 0 : 1 Þð 0 : 1 Þð 0 : 1 Þð 0 : 1 Þð 1 
 0 : 1 Þ¼ð 0 : 1 Þ^4 ð 0 : 9 Þ

The probability of obtaining all fiveS’s is

ð 0 : 1 Þð 0 : 1 Þð 0 : 1 Þð 0 : 1 Þð 0 : 1 Þ¼ð 0 : 1 Þ^5

Since the event ‘‘all five with side e¤ects’’ corresponds to only one of the 32
outcomes above and the event ‘‘four with side e¤ects and one without’’ pertains
to five of the 32 outcomes, each with probabilityð 0 : 1 Þ^4 ð 0 : 9 Þ, the addition rule
yields a probability

ð 0 : 1 Þ^5 þð 5 Þð 0 : 1 Þ^4 ð 0 : 9 Þ¼ 0 : 00046

for the compound event that ‘‘four or more have side e¤ects.’’ In general, the
binomial model applies when each trial of an experiment has two possible out-
comes (often referred to as ‘‘failure’’ and ‘‘success’’ or ‘‘negative’’ and ‘‘posi-
tive’’; one has a success when the primary outcome is observed). Let the prob-
abilities of failure and success be, respectively, 1
pandp, and we ‘‘code’’
these two outcomes as 0 (zero successes) and 1 (one success). The experiment
consists ofnrepeated trials satisfying these assumptions:

1. Thentrials are all independent.


2. The parameterpis the same for each trial.

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