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### Binomial Distribution

Sometimes 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)= 1-1/6 =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) = nCx.px.qn-x.

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) = 6Cx.1/2x.1/26-x.
P(x) = 6Cx.1/26.
Case a Now consider the first case. When exactly three heads are obtained x= 3.
P(3) = 6C3.1/26 = 5/16
Case b When exactly 4 heads are obtained, x = 4 or 5 or 6. P(4)+P(5)+P(6) = 6C4.1/26 + 6C5.1/26 + 6C6.1/26.
= 1/26.[6C4 + 6C5 + 6C6]
= 11/32
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