Probability
Hypergeometric Distribution — Variance
nta_pyq_2024_apr
Grade 12
Question:
From a lot of 12 items containing 3 defectives, a sample of 5 items is drawn at random. Let the random variable $X$ 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 $\gcd(m,n)=1$, then $n-m$ is equal to
Step-by-Step Solution
Key Concept: Hypergeometric distribution: $X\sim\text{Hyp}(N=12,K=3,n=5)$. $\sigma^2=n\cdot\frac{K}{N}\cdot\frac{N-K}{N}\cdot\frac{N-n}{N-1}=5\cdot\frac{3}{12}\cdot\frac{9}{12}\cdot\frac{7}{11}$.
$\sigma^2=105/176$. $n-m=71$.
Correct Answer: 71