Binomial Variable

Binomial variables are frequently encountered in epidemiological data, and the binomial distribution is used to model the prevalence rate and cumulative incidence rate. Binomial variables are created through repeated trials of a Bernoulli process, often called Bernoulli trials. Daniel Bernoulli (1700–1782) was the first mathematician to work out the mathematics for Bernoulli trials. Bernoulli trials must satisfy the following three conditions:

• The experiment has two possible outcomes, labeled success (S) and failure (F).
• The trials are independent.
• The probability of a success remains the same from trial to trial; this is called the success probability and is denoted with the letter p.

The word success as used here is arbitrary and does not necessarily represent something good. Either of the two possible categories may be called the success S as long as the corresponding probability is identified as p.

A random variable for the number of successes in a sequence of Bernoulli trials is called a binomial variable. The probability distribution for a binomial variable is called the binomial distribution. The binomial probability formula for the number of successes, X, is

where the binomial coefficient (n/k) is defined as

and k! is the product of the first k positive integers and is called k factorial. In symbols,

Consider the following example. Epidemiological surveys have determined that 9% of men and 0.25% of women cannot distinguish between the colors red and green. This is the type of color blindness that causes problems reading traffic signals. If six men are randomly selected for a study of traffic signal perceptions, the probability that exactly two of them cannot distinguish between red and green can be calculated using the binomial distribution formula as follows.

We must carefully define which outcome we wish to call a success. For convenience, we define a success as a randomly selected man who cannot distinguish between the colors red and green, so p = 0.09. Let X denote the number of men of the six who cannot distinguish between the colors red and green. The number of trials is the number of men in the study, so that n = 6. Using the binomial probability formula for k = 2 yields

Therefore, the probability that exactly two of the six men cannot distinguish between red and green is 8.33%.