Two cards are drawn successively with replacement from fair playing 52 cards. let X denote number of kings…

Let $X$ represent the number of kings obtained when drawing two cards with replacement from a standard 52-card deck. The probability of drawing a king is $P(K) = \frac{4}{52} = \frac{1}{13}$, while the probability of not drawing a king is $P(K') = \frac{12}{13}$.

Since the draws are independent, the probability distribution of $X$ is given by:

$P(X=0) = P(K') \times P(K') = \frac{12}{13} \cdot \frac{12}{13} = \frac{144}{169}$

$P(X=1) = P(K)P(K') + P(K')P(K) = 2 \cdot \frac{1}{13} \cdot \frac{12}{13} = \frac{24}{169}$

$P(X=2) = P(K) \times P(K) = \frac{1}{13} \cdot \frac{1}{13} = \frac{1}{169}$

These probabilities sum to $\frac{144+24+1}{169} = 1$, confirming a valid distribution.

The expected value of $X^2$ is calculated as:

$E(X^2) = \sum x^2 P(X=x) = 0^2 \cdot \frac{144}{169} + 1^2 \cdot \frac{24}{169} + 2^2 \cdot \frac{1}{169}$

$E(X^2) = 0 + \frac{24}{169} + \frac{4}{169} = \frac{28}{169}$

The expected value of $X^2$ is $\frac{28}{169}$, corresponding to choice D.