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
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