Question

If $X$ follows binomial distribution with parameters $n=5$, $p$ and $P(X=2)=9$, $P(X=3)$ , then $p=$

Answer 1

The binomial distribution formula for any random variable $X$ is given by
$P(X=r)={}^n{C}_r{p}^r{q}^{n-r}$
where,
$n=\text{the number of experiments}$
$x=0,1,2,3,4,\dots$
$p=\text{Probability of success in a single experiment}$
and $q=\text{Probability of failure in a single experiment}(=1–p)$
Given, $P(X=2)=9\cdot P(X=2)$
$\Rightarrow {}^5{C}_2{p}^2{(1-p)}^3$
$=9\cdot {}^5{C}_2{p}^3{(1-p)}^2$ $(\because p+q=1)$
$\Rightarrow \frac{1-p}{p}=9$ $[\because {}^n{C}_r={}^n{C}_{n-r}]$
$\Rightarrow p=\frac{1}{10}$
Hence, answer is $\frac{1}{10}$.