Let $\mathrm{X}$ be random variable having Binomial distribution $\mathrm{B}(7, \mathrm{p})$. If…

Let $\mathrm{X}$ be random variable having Binomial distribution $\mathrm{B}(7, \mathrm{p})$. If $\mathrm{P}[\mathrm{X}=3]=5 \mathrm{P}[\mathrm{X}=4]$, then variance of $\mathrm{X}$ is
  1. $\frac{7}{6}$
  2. $\frac{35}{36}$
  3. $\frac{77}{36}$
  4. $\frac{1}{36}$

Solution

$\begin{aligned} & \mathrm{P}(\mathrm{X}=3)=5 \mathrm{P}(\mathrm{X}=4) \\ & \Rightarrow{ }^7 \mathrm{C}_3 \mathrm{p}^3 \mathrm{q}^4=5^7 \mathrm{C}_4 \mathrm{p}^4 \mathrm{q}^3 \\ & \Rightarrow 5 \mathrm{p}=\mathrm{q} \\ & \Rightarrow 5 \mathrm{p}=1-\mathrm{p} \\ & \Rightarrow 6 \mathrm{p}=1 \\ & \Rightarrow \mathrm{p}=\frac{1}{6} \\ & \Rightarrow \mathrm{q}=1-\frac{1}{6}=\frac{5}{6} \\ & \text { Variance }=\mathrm{npq} \\ & =7 \times \frac{1}{6} \times \frac{5}{6} \\ & =\frac{35}{36}\end{aligned}$

Asked in: MHT CET 2023 (13 May Shift 1)

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