Question

Two cards are drawn successively with replacement from fair playing 52 cards. let X denote number of kings obtained when two cards are drawn, then \(\mathrm{E}\left(\mathrm{X}^2\right)=\)

• A. \(\frac{24}{169}\)
• B. \(\frac{26}{169}\)
• C. \(\frac{27}{169}\)
• D. \(\frac{28}{169}\)

Hint:

For replacement, probabilities remain constant for each draw.

Answer 1

The Correct Option is D

Step 1: Probability Distribution
This is a Binomial distribution with (n=2, p = \frac{4}{52} = \frac{1}{13}, q = \frac{12}{13}).
(P(X=0) = q^2 = \frac{144}{169})
(P(X=1) = 2pq = 2(\frac{1}{13})(\frac{12}{13}) = \frac{24}{169})
(P(X=2) = p^2 = \frac{1}{169}).

Step 2: Calculate Expectation
(E(X^2) = \sum x^2 P(x))
(E(X^2) = (0^2 \cdot \frac{144}{169}) + (1^2 \cdot \frac{24}{169}) + (2^2 \cdot \frac{1}{169})).
(E(X^2) = \frac{24}{169} + \frac{4}{169} = \frac{28}{169}).
Final Answer: (D)