The binomial distribution describes the number of times that a particular event will occur in a sequence of observations. The event is coded binary, assuming values zero and one. The binomial distribution is used when a researcher is interested in the occurrence of an event. For instance, in a clinical trial, a patient may survive or die. The researcher studies the number of survivors, and not how long the patient survives after treatment.
The binomial distribution is specified by the number of observations,denoted by n, and the probability of occurence, which is denoted by p.
A classic example that is used often to illustrate concepts of probability theory, is the tossing of a coin. If a coin is tossed 4 times, then we may obtain 0, 1, 2, 3, or 4 heads. We may also obtain 4, 3, 2, 1, or 0 tails, but these outcomes are equivalent to 0, 1, 2, 3, or 4 heads. The likelihood of obtaining 0, 1, 2, 3, or 4 heads is, respectively, 1/16, 4/16, 6/16, 4/16, and 1/16. In the figure on this page the distribution is shown with p = 1/2 Here in this example, one is likely to obtain 2 heads in 4 tosses, since this outcome has the highest probability.
Other situations in which binomial distributions arise are quality control, public opinion surveys, medical research, and insurance problems.