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).
here the i= number of times our desired outcome has occurred and i=0,1,2.....n
four coins are tossed what is the probability of getting two head and two tail?
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