Five bad apples are mixed with 15 good apples accidentally. Let the random variable \( x \) be the number of bad apples in a draw of two apples, then mean of \( x \) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( \displaystyle\frac{1}{2} \)
There are 20 apples, so two apples can be drawn in \( ^{20}C_{2}=190 \) ways. X can take the values 0, 1, 2:
\( \displaystyle P(X=0)=\frac{^{15}C_{2}}{190}=\frac{105}{190} \)
\( \displaystyle P(X=1)=\frac{5\times 15}{190}=\frac{75}{190} \)
\( \displaystyle P(X=2)=\frac{^{5}C_{2}}{190}=\frac{10}{190} \)
The mean is
\( \displaystyle E(X)=0\cdot\frac{105}{190}+1\cdot\frac{75}{190}+2\cdot\frac{10}{190} \)
\( \displaystyle =\frac{95}{190}=\frac{1}{2} \)
Shortcut: Each apple drawn is bad with probability \( \frac{5}{20}=\frac{1}{4} \), and the expected number of bad apples in two draws is \( 2\times\frac{1}{4}=\frac{1}{2} \), with or without replacement.
CBSE Applied Mathematics (Class XII):
Mathematical Expectation of Discrete Probability Distribution
