Question

For a binomial variate X, if n = 3 and P (X = 1) = 8 P (X = 3), then p =
(a) 4/5
(b) 1/5
(c) 1/3
(d) 2/3

Answer 1

n =3

$$\begin{gathered} P(X=1)=8P(X=3)\left(\mathrm{Given}\right) \\ \text{The d}\mathrm{istribution}\mathrm{is}\mathrm{given}\mathrm{by} \\ P(X=r)={}^{3}C_r{\left(p\right)}^r{\left(q\right)}^{3\mathit{-}r} \\ P\mathit{(}X=1)\mathit{=}{}^{\mathit{3}}C_1{\left(p\right)}^1{\left(q\right)}^2\mathit{}\mathrm{and}P\mathit{(}X=3)\mathit{=}{}^{3}C_3{\left(p\right)}^3{\left(q\right)}^0 \\ \Rightarrow 3p{q}^2=8p^3 \\ \Rightarrow 8p^{\mathit{2}}=3q^2 \\ \Rightarrow 8p^{\mathit{2}}=3(1-\mathrm{p}{)}^2 \\ \Rightarrow 8p^{\mathit{2}}=3-6\mathrm{p}+3{\mathrm{p}}^2 \\ \Rightarrow 5{\mathrm{p}}^{\mathit{2}}+6\mathrm{p}-3=0 \\ \Rightarrow p=\frac{-6\pm \sqrt{96}}{10} \end{gathered}$$

Hence , it does not match any of the answer choices.