Question

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

Answer 1

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