Binomial Random Variable Introduction with an example

Informally if I have to define Binomial Random Variable, I will define It as the sum of all the value of Bernoulli Random Variable. or collection of Bernoulli experiment.

Suppose that n independent trials, each of which results in a “success” with probability p and in a “failure” with probability 1 − p, are to be performed. If X
represents the number of successes that occur in the n trials, then X is said to be
a binomial random variable with parameters (n,p).

P(i) = \binom{n}{i}(p)^{^{i}}(1-p)^{n-i}

here the i= number of times our desired outcome has occurred and                                        i=0,1,2.....n

Example:-

four coins are tossed what is the probability of getting two head and two tail?

Solution:-

here i=2, n=4, and 

probability of getting head or tail  =1/2

So p= 1/2 ,  1-p= 1/2 

putting on the formula 

   P(i=2) =  (4 c 2) (1/2)2 (1/2)2 =3/8

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