Probability
Probability
nta_pyq_2025_jan
Grade 12
Question:
Three defective oranges are accidentally mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If $X$ denotes the number of defective oranges, then the variance of $X$ is:
$\dfrac{28}{75}$
$\dfrac{18}{25}$
$\dfrac{26}{75}$
$\dfrac{14}{25}$
Step-by-Step Solution
Key Concept: Hypergeometric: $X\sim\text{Hyp}(N=10,K=3,n=2).$ $P(X=k)=\dfrac{\binom{3}{k}\binom{7}{2-k}}{\binom{10}{2}}.$ Compute $E[X]$ and $E[X^{2}]$ directly.
$P(X=0)=\dfrac{\binom{7}{2}}{\binom{10}{2}}=\dfrac{21}{45}=\dfrac{7}{15}.$
$P(X=1)=\dfrac{\binom{3}{1}\binom{7}{1}}{45}=\dfrac{21}{45}=\dfrac{7}{15}.$
$P(X=2)=\dfrac{\binom{3}{2}}{45}=\dfrac{3}{45}=\dfrac{1}{15}.$
$E[X]=0+\dfrac{7}{15}+\dfrac{2}{15}=\dfrac{9}{15}=\dfrac{3}{5}.$
$E[X^{2}]=0+\dfrac{7}{15}+\dfrac{4}{15}=\dfrac{11}{15}.$
$\operatorname{Var}(X)=\dfrac{11}{15}-\dfrac{9}{25}=\dfrac{55-27}{75}=\dfrac{28}{75}.$
Correct Answer: 1