Let a random variable $X$ has a binomial distribution with mean $8$ and variance 4. If $P\left(X\le 2\right)=\frac{k}{2^{16}},$ then the value of $k$ is equal to

Let $n$ be the number of times the experiment is done, $p$ is the probability of the success and $q$ is the probability of the failure.

$\therefore$ $np=8$ (mean), $npq=4$ (variance)

$\Rightarrow q=\frac{1}{2}\Rightarrow p=\frac{1}{2}$

$\Rightarrow n=16$

$\Rightarrow P\left(X\le 2\right)=p\left(X=0\right)+p\left(X=1\right)+p\left(X=2\right)$

$\Rightarrow P\left(X\le 2\right){=}^{16}{C}_0{p}^0{q}^{16}{+}^{16}{C}_1{p}^1{q}^{15}{+}^{16}{C}_2{p}^2{q}^{14}$

$\Rightarrow P\left(X\le 2\right)={\left(\frac{1}{2}\right)}^{16}+16{\left(\frac{1}{2}\right)}^{16}+120\times {\left(\frac{1}{2}\right)}^{16}$

$\Rightarrow P\left(X\le 2\right)={\left(\frac{1}{2}\right)}^{16}\left(137\right)=\frac{137}{2^{16}}=\frac{k}{2^{16}}$

$\therefore k=137$