Many random experiments include counting the number of successes in a series of a fixed number of independently repeated trials, which may result in either success or failure. The distribution of the number of successes is a binomial distribution. It is a discrete probability distribution with two parameters, traditionally indicated by n, the number of trials, and p, the probability of success. Such a success/failure experiment is also called a Bernoulli experiment, or Bernoulli trial; when n = 1, the Bernoulli distribution is a binomial distribution.
Named after Jacob Bernoulli, who studied them extensively in the 1600s, a well known example of such an experiment is the repeated tossing of a coin and counting the number of times "heads" comes up.
In a sequence of Bernoulli trials, we are often interested in the total number of successes and not in the order of their occurrence. If we let the random variable X equal the number of observed successes in n Bernoulli trials, the possible values of X are 0,1,2,…,n. If x success occur, where x=0,1,2,...,n , then n-x failures occur. The number of ways of selecting x positions for the x successes in the x trials is:
Since the trials are independent and since the probabilities of success and failure on each trial are, respectively, p and q=1−p, the probability of each of these ways is px(1−p)n−x. Thus, the p.d.f. of X, say f(x) , is the sum of the probabilities of these (nx) mutually exclusive events--that is,
These probabilities are called binomial probabilities, and the random variable X is said to have a binomial distribution .