Probability
Variance of Hypergeometric Distribution
nta_pyq_2024_jan
Grade 12
Question:
Three rotten apples are accidentally mixed with fifteen good apples. Assuming the random variable $x$ to be the number of rotten apples in a draw of two apples, the variance of $x$ is
$\dfrac{37}{153}$
$\dfrac{57}{153}$
$\dfrac{47}{153}$
$\dfrac{40}{153}$
Step-by-Step Solution
Key Concept: Hypergeometric: 18 total (3 rotten, 15 good), draw 2. $P(X=0)=\frac{{}^{15}C_2}{{}^{18}C_2}=\frac{105}{153}$, $P(X=1)=\frac{{}^3C_1\cdot{}^{15}C_1}{{}^{18}C_2}=\frac{45}{153}$, $P(X=2)=\frac{{}^3C_2}{{}^{18}C_2}=\frac{3}{153}$.
$\text{Var}(X)=E(X^2)-(E(X))^2=57/153-1/9=40/153$.
Correct Answer: 4