Question

Three rotten apples are accidently 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

• A. \(\frac{37}{153}\)
• B. \(\frac{57}{153}\)
• C. \(\frac{47}{153}\)
• D. \(\frac{40}{153}\)

Answer 1

The Correct Option is D

To determine the variance of the random variable \(x\), representing the number of rotten apples drawn from a selection of two apples from a mixed batch, follow these computations:

Step 1: Compute probabilities.

• Total apples = 3 rotten + 15 good = 18 apples.
• The total number of combinations for selecting 2 apples from 18 is \(\binom{18}{2} = \frac{18 \times 17}{2} = 153\).

Step 2: Define the probability distribution for \(x\).

• \(P(x = 0)\): This is the probability of selecting 0 rotten apples (meaning 2 good ones). Calculated as selecting 2 from 15 good apples: \(P(x=0)= \frac{\binom{15}{2}}{\binom{18}{2}} = \frac{105}{153}\)
• \(P(x = 1)\): This is the probability of selecting 1 rotten and 1 good apple. Calculated as: \(P(x=1) = \frac{\binom{3}{1} \times \binom{15}{1}}{\binom{18}{2}} = \frac{45}{153}\)
• \(P(x = 2)\): This is the probability of selecting 2 rotten apples. Calculated as selecting 2 from the 3 rotten apples: \(P(x=2) = \frac{\binom{3}{2}}{\binom{18}{2}} = \frac{3}{153}\)

Step 3: Compute the expected value \(E(x)\).

• The formula for \(E(x)\) is \(E(x) = \sum x P(x)\).
• \(E(x) = 0 \times \frac{105}{153} + 1 \times \frac{45}{153} + 2 \times \frac{3}{153} = \frac{51}{153}\)

Step 4: Compute the expected value of \(x^2\), \(E(x^2)\).

• The formula for \(E(x^2)\) is \(E(x^2) = \sum x^2 P(x)\).
• \(E(x^2) = 0^2 \times \frac{105}{153} + 1^2 \times \frac{45}{153} + 2^2 \times \frac{3}{153} = \frac{57}{153}\)

Step 5: Compute the variance \(Var(x)\).

• The variance formula is \(Var(x) = E(x^2) - [E(x)]^2\).
• \(Var(x) = \frac{57}{153} - \left(\frac{51}{153}\right)^2 = \frac{57}{153} - \frac{2601}{23409} = \frac{6120}{23409} - \frac{2601}{23409} = \frac{3519}{23409} = \frac{40}{153}\)

The variance of \(x\) is thus \( \frac{40}{153} \).