Question

If X follows binomial distribution with parameters n = 5, p and P(X = 2) = 9P(X = 3), then find the value of p. [CBSE 2015]

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ X\mathrm{follows}\mathrm{binomial}\mathrm{distribution}\mathrm{with}\mathrm{parameters}n=5,p\mathrm{and}\mathrm{P}\left(X=2\right)=9\mathrm{P}\left(X=3\right). \\ \mathrm{So},\mathrm{P}\left(\mathrm{X}=r\right)={}^5\mathrm{C}_r{p}^r{q}^{\left(5-r\right)},\mathrm{where}r=0,1,2,3,4,5\mathrm{and}q=1-p \\ \mathrm{As},\mathrm{P}\left(X=2\right)=9\mathrm{P}\left(X=3\right) \\ \Rightarrow {}^5\mathrm{C}_2{p}^2{q}^3=9{}^5\mathrm{C}_3{p}^3{q}^2 \\ \Rightarrow 10p^2{q}^3=9\times 10p^3{q}^2 \\ \Rightarrow q=9p \\ \Rightarrow 1-p=9p\left[\mathrm{As},q=1-p\right] \\ \Rightarrow 10p=1 \\ \therefore p=\frac{1}{10} \end{gathered}$$