Question

The mean and variance of a random variable $X$ having a binomial distribution are $6$ and $3$ respectively. Find the probability of variable $X$ less than $2$.

Answer 1

We know that, mean and variance of binomial distribution are given by $np=6$ and $npq=3$ respectively.
Solving, we get
$q=\frac{1}{2},p=\frac{1}{2},n=12$

Therefore, probability of variable $X$ less than $2$ is given by
$P(X=0)+P(X=1)={}^{12}{C}_0{p}^0{q}^{12}+{}^{12}{C}_1{p}^1{q}^{11}$
$={\left(\frac{1}{2}\right)}^{12}+12\times {\left(\frac{1}{2}\right)}^1{\left(\frac{1}{2}\right)}^{11}$
$=\frac{13}{2^{12}}$
$=\frac{13}{4096}$