Binomial random variable is a discrete random variable which is related to Binomial experiments or Bernoulli trials. Experiments whose outcomes can be reduced to two outcomes as success and failure are called Binomial experiments. For example, a multiple choice question can have five options. But the outcome of attempting the question turns out to be either right or wrong, which can be treated as success and failure. If there are 10 such questions in a test and a student just selects the options guessing, then the Binomial random variable here can denote the number of questions which he got right.

A binomial random variable X is the number of successes in n independent trials of a Binomial experiment and is denoted by B(n, p) where p is the probability of success in a single trial.
Binomial Probability Formula:

In a Binomial experiment the probability of exactly X successes in n trials is
P(X) = C(n, X).pX.qn-X

where
q = 1 - p (the probability of failure).

X - number of successes

n - number of trials

n-X - number of failures

p - probability of success in one trial

The values a binomial random variable assumes are tabulated against the respective probabilities, which is generally called a Binomial distribution.

Consider the situation in a three child family. The sample space for the sex of the children is
S = {BBB, BBG, BGB, GBB, BGG, GBG, GGB, GGG}  where B and G denote a boy and a girl respectively.

Suppose the Binomial random variable X denotes the number of girls in the family,
X = {0, 1, 2, 3}.
p = probability of a child born being a girl = $\frac{1}{2}$.

q = 1 - p = $\frac{1}{2}$.

Using the Binomial probability formula, the probabilities of P(X = 0), P(X = 1), P(X = 2) and P(X = 3) can be calculated.

The Binomial distribution is shown in the table below:

   X
  
  1 
    
 P(X) 
 $\frac{1}{8}$ $\frac{3}{8}$
$\frac{3}{8}$ $\frac{1}{8}$
  
Suppose X and Y are two independent binomial random variables with the same probability of success p. If the numbers of trials for X and Y are n and m, then the sum X + Y is also a Binomial variable B(m + n, p) with the same probability of success.

Solved Example

Question: A die is thrown 6 times. If turning a six is considered as a success, find the probability of getting
1. at least 4 successes
2. no success.
Solution:
 
We calculate the binomial probabilities using p = $\frac{1}{6}$, q = $\frac{5}{6}$ and n = 6

1. P(X≥4) = P(4) + P(5) + P(6)

                = C(6, 4)($\frac{1}{6}$)4 ($\frac{5}{6}$)2 + C(6, 5)($\frac{1}{6}$)5($\frac{1}{6}$)1 + C(6, 6)($\frac{1}{6}$)6($\frac{5}{6}$)0.

                ≈ 0.00804 + 0.00013 + 0.00002 = 0.00819

2. P(X = 0) = C(6, 0)($\frac{1}{6}$)0($\frac{5}{6}$)6.   ≈ 0.33490.