Consider a random experiment whose outcomes can be classified as success or failure. It means that experiment results in only two outcomes E1(success) or E2­ (failure). Further assume that experiment can be repeated several times, probability of success or failure in any trial are p and q (p + q = 1) and don’t vary from trial to trial and finally different trials are independent. Such a experiment is called Binomial experiment and trials are said to be binomial trials. For instance tossing of a fair coin several times, each time outcome would be either a success (say occurrence of head) or failure (say occurrence of tail).

A probability distribution representing the binomial trials is said to binomial distribution.

Let us consider a Binomial experiment which has been repeated ‘n’ times. Let the probability of success and failure in any trial be p and q respectively and we are interested in the probability of occurrence of exactly ‘r’ successes in these n trials. Now number of ways of choosing ‘r’ success in ‘n’ trials = nCr. Probability of ‘r’ successes and (n-r) failures is pr×qn-r. Thus probability of having exactly r successes =  nCr×pr×qn-r

Let ‘X’ be random variable representing the number of successes, then

P(X = r) = nCr×pr×qn-r   (r = 0, 1, 2, L , n)

1 = (p + q)n = nC0p0qn + nC1p1qn –1 + nC2p2qn –2 + ……. + nCrprqn –r + ……. + nCnpn

X ->               Number of success

0,              1,                2,                  …….., r,        ………..         n 