JEE Main 2024 (Online) 5th April Morning ShiftQuestion 1
From a lot of $10$ items, which include $3$ defective items, a sample of $5$ items is drawn at random. Let the random variable $X$ denote the number of defective items in the sample. If the variance of $X$ is $\sigma^{2}$, then $96\sigma^{2}$ is equal to $\_\_\_\_\_\_\_\_$ .
Probability
Answer$56$
Solution
$X$ is hypergeometric with $N=10$, $K=3$, $n=5$: $\sigma^2=5\cdot\frac3{10}\cdot\frac7{10}\cdot\frac{5}{9}=\frac7{12}$, so $96\sigma^2=56$.