JEE Main · 2024 · 06 Apr 2024 · Shift 2 · Probability

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

\frac{m}{n}

where

g cd (m, n) = 1

then

n - m

is equal to ______.

Answer: 71

Solution

For sampling without replacement from 12 items with 3 defective, 5 drawn, the hypergeometric variance of the number of defectives is

Var (X) = n \cdot \frac{K}{N} (1 - \frac{K}{N}) \cdot \frac{N - n}{N - 1} = 5 \cdot \frac{3}{12} \cdot \frac{9}{12} \cdot \frac{7}{11} = \frac{105}{176}

With

g cd (105, 176) = 1

m = 105, n = 176

, so

n - m = 71

JEE Main · 2024 · 06 Apr 2024 · Shift 2 · Integer answer