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 _______.

Answer 1

Correct answer is 71

\(\mathrm{a}=1-\frac{{ }^{3} \mathrm{C}_{5}}{{ }^{12} \mathrm{C}_{5}}\)

\(\mathrm{b}=3 \cdot \frac{{ }^{9} \mathrm{C}_{4}}{{ }^{12} \mathrm{C}_{5}}\)

\(\mathrm{c}=3 \cdot \frac{{ }^{9} \mathrm{C}_{3}}{{ }^{12} \mathrm{C}_{5}}\)

\(\mathrm{d}=1 \cdot \frac{{ }^{9} \mathrm{C}_{2}}{{ }^{12} \mathrm{C}_{5}}\)

\(\mathrm{u}=0 . \mathrm{a}+1 . \mathrm{b}+2 . \mathrm{c}+3 . \mathrm{d}=1.25\)

\(\sigma^{2}=0 . a+1 . b+4 . c+9 d-u^{2}\)

\(\sigma^{2}=\frac{105}{176}\)

\(176-105=71\)