Bernoulli Trials and Binomial Distribution

Bernoulli trials:
The outcome of any trial is independent of the outcome of any other trial. In each of such trials, the probability of success or failure remains constant. Such independent trials which have only two outcomes usually referred as ‘success’ or ‘failure’ are called Bernoulli trials. 

Definition:  
Trials of a random experiment are called Bernoulli trials, if they satisfy the following conditions : 
(i) There should be a finite number of trials. 
(ii) The trials should be independent. 
(iii) Each trial has exactly two outcomes : success or failure.
(iv) The probability of  success remains the same in each trial. 
For example, throwing a die 50 times is a case of 50 Bernoulli trials, in which each trial results in success (say an even number) or failure (an odd number) and the probability of success (p) is same for all 50 throws. Obviously, the successive throws of the die are independent experiments. If the die is fair and have six numbers 1 to 6 written on six faces, then  
p = `1/2 and q = 1 -p =1/2` probability of failure.

Binomial distribution:

Let us take the experiment made up of three Bernoulli trials with probabilities p and q = 1 – p for success and failure respectively in each trial. The sample space of the experiment is the set 
S = {SSS, SSF, SFS, FSS, SFF, FSF, FFS, FFF} 
The number of successes is a random variable X and can take values 0, 1, 2, or 3. The probability distribution of the number of successes is as below : 
P(X = 0) = P(no success) 
= P({FFF}) = P(F) P(F) P(F) 
= `q . q . q = q^3` since  the trials are independent 

P(X = 1) = P(one successes) 
= P({SFF, FSF, FFS}) 
= P({SFF}) + P({FSF}) + P({FFS}) 
= P(S) P(F) P(F) + P(F) P(S) P(F) + P(F) P(F) P(S) 
= `p.q.q + q.p.q + q.q.p = 3pq^2 `

P(X = 2) = P (two successes)
= P({SSF, SFS, FSS})
= P({SSF}) + P ({SFS}) + P({FSS})
= P(S) P(S) P(F) + P(S) P(F) P(S) + P(F) P(S) P(S) 
= `p.p.q. +  p.q.p  +  q.p.p  =  3p^2q` 
and
P(X = 3) = P(three success) = P ({SSS}) 
= P(S) . P(S) . P(S) = `p^3` 
Thus, the probability distribution of X is 

Also, the binominal expansion of `(q + p)^3` is
`q^3 + 3q^2p + 3 qp^2 + p^3`
Thus, we may conclude that in an experiment of n-Bernoulli trials, the probabilities of 0, 1, 2,..., n successes can be obtained as 1st, 2nd,...,`(n + 1)^(th)` terms in the expansion of `(q + p)^n`.
The probability of x successes in n-Bernoulli trials is `(n!)/ (x!(n - x)!) p^x q^(n -x)`

or `"^nC_x  p^x q^(n-x)`

The probability of x successes P(X = x) is also denoted by P(x) and is given by 

P(x) = `"^nC_x q^(n-x) p^x` ,  x = 0, 1,..., n. (q = 1 – p) 
This P(x) is called the probability function of the binomial distribution.