For a Binomial distribution, $n=6$, if $9 P(X=4)=P(X=2)$, then $q=$

For a Binomial distribution, $n=6$, if $9 P(X=4)=P(X=2)$, then $q=$
  1. $\frac{1}{2}$
  2. $\frac{1}{4}$
  3. $\frac{3}{4}$
  4. $\frac{2}{5}$

Solution

$\begin{aligned} & 9 P(X=4)=P(X=2) \\ & \Rightarrow 9{ }^n C_4 P^4 \cdot q^{n-4}={ }^n C_2 P^2 q^{n-2} \\ & \Rightarrow 9 \frac{{ }^n C_4}{{ }^n C_2}=\frac{q^2}{P^2}=\left(\frac{q}{1-q}\right)^2 \\ & \Rightarrow \frac{q}{1-q}=3 \Rightarrow q=\frac{3}{4}\end{aligned}$

Asked in: MHT CET 2022 (07 Aug Shift 1)

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