
Binomial DistributionSometimes we may be interested in the happening of a certain event A of S and the 2 outcomes can be :(a) A happens (b)A does not happen For eg:If a die is tossed and and we are interested in the finding that A = {5} .The occurence of A is called 'success' and can be denoted by S. So P(S)=1/6 and P(F)= 11/6 =5/6. Such trials with 2 outcomes are called Bernoulli Trials. Let us perform an action involving repetition of Bernoulli trials. Let the constant probability of success be p and that of failure be q. Then the probability of getting x successes from n independent trials is given by P(x) = ^{n}C_{x}.p^{x}.q^{nx}. Solved Example An unbiased coin is tossed 6 times. Find the probability of getting (a)3 heads,(b)at least 4 heads. Solution 6 independent trials are carried out . So n = 6 The probability of success, i.e getting a head = p = 1/2. The probability of failure, i.e getting a tail = q = 1/2. P(x) = ^{6}C_{x}.1/2^{x}.1/2^{6x}. P(x) = ^{6}C_{x}.1/2^{6}. Case a Now consider the first case. When exactly three heads are obtained x= 3. P(3) = ^{6}C_{3}.1/2^{6} = 5/16 Case b When exactly 4 heads are obtained, x = 4 or 5 or 6. P(4)+P(5)+P(6) = ^{6}C_{4}.1/2^{6} + ^{6}C_{5}.1/2^{6} + ^{6}C_{6}.1/2^{6}. = 1/2^{6}.[^{6}C_{4} + ^{6}C_{5} + ^{6}C_{6}] = 11/32 