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Consider a random experiment whose outcomes can be classified as success or failure. It means that experiment results in only two outcomes E_{1}(success) or E_{2} (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 = ^{n}C_{r}. Probability of ‘r’ successes and (n-r) failures is p^{r}×q^{n}^{-r}. Thus probability of having exactly r successes = ^{n}C_{r}×p^{r}×q^{n}^{-r}

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

P(X = r) = ^{n}C_{r}×p^{r}×q^{n}^{-r} (r = 0, 1, 2, L , n)

1 = (p + q)^{n} = ^{n}C_{0}p^{0}q^{n} + ^{n}C_{1}p^{1}q^{n –1} + ^{n}C_{2}p^{2}q^{n –2} + ……. + ^{n}C_{r}p^{r}q^{n –r} + ……. + ^{n}C_{n}p^{n}

X -> Number of success

0, 1, 2, …….., r, ……….. n

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