Question

The mean and variance of a binomial distribution are $10$ and $\frac{5}{3}$ respectively. Find $P(x\ge 1)$.

Answer 1

Given mean and variance of a binomial distribution are

10 and $\frac{5}{3}$ respectively.

Let the parametrs be n,p,q

$\Rightarrow np=10,npq=\frac{5}{3}$

$\Rightarrow q=\frac{5}{\frac{3}{10}}=\frac{1}{6}$

$p=1-q=1-\frac{1}{6}=\frac{5}{6}$

$np=10\Rightarrow n.\frac{5}{6}=10\Rightarrow n=12$

$p(x\ge 1)=2^n-p(x=0)$

$=2^{12}{-}^{12}{C}_0{\left(\frac{5}{6}\right)}^0{\left(\frac{1}{6}\right)}^{12}$

$=2^{12}-{\left(\frac{1}{6}\right)}^{12}$