In statistics the so-called binomial distribution describes the possible number of times that a particular event will occur in a sequence of observations. The event is coded binary, it may or may not occur. The binomial distribution is used when a researcher is interested in the occurrence of an event, not in its magnitude. 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. Another example is whether a person is ambitious or not. Here, the binomial distribution describes the number of ambitious persons, and not how ambitious they are.
The binomial distribution is specified by the number of observations, 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 Thus, in the example discussed here, 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.
If the probability p is small and the number of observations is large the binomial probabilities are hard to calculate. In this instance it is much easier to approximate the binomial probabilities by poisson probabilities. The binomial distribution approaches the poisson distribution for large n and small p. In the movie we increase the number of observations from 6 to 50, where the parameter p in the binomial distribution remains 1/10. The movie shows that the degree of approximations improves as the number of observations increases.
For moderate values of p, the binomial distribution approaches the normal distribution if the number of observations is large. This is an example of the central limit theorem.