From a lot of 12 items containing 3 defectives, a sample of 5 items is drawn at random. Let the random variable $\mathrm{X}$ denote the number of defective items in the sample. Let items in the sample be drawn one by one without replacement. If variance of $X$ is $\frac{m}{n}$, where…

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From a lot of 12 items containing 3 defectives, a sample of 5 items is drawn at random. Let the random variable $\mathrm{X}$ denote the number of defective items in the sample. Let items in the sample be drawn one by one without replacement. If variance of $X$ is $\frac{m}{n}$, where $\operatorname{gcd}(m, n)=1$, then $n-m$ is equal to _________

Aicharya · Accepted Answer

$\begin{aligned} & \mathrm{a}=1-\frac{{ }^3 \mathrm{C}_5}{{ }^{12} \mathrm{C}_5} \\ & \mathrm{~b}=3 \cdot \frac{{ }^9 \mathrm{C}_4}{{ }^{12} \mathrm{C}_5} \\ & \mathrm{c}=3 \cdot \frac{{ }^9 \mathrm{C}_3}{{ }^{12} \mathrm{C}_5} \\ & \mathrm{~d}=1 \cdot \frac{{ }^9 \mathrm{C}_2}{{ }^{12} \mathrm{C}_5} \\ & \mathrm{u}=0 . \mathrm{a}+1 . b+2 \cdot c+3 \cdot \mathrm{d}=1.25 \\ & \sigma^2=0 . \mathrm{a}+1 . b+4 . c+9 \mathrm{~d}-\mathrm{u}^2 \\ & \sigma^2=\frac{105}{176} \end{aligned}$
Ans. $176-105=71$

From a lot of 12 items containing 3 defectives, a sample of 5 items is drawn at random. Let the random…

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