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(A) Let $X$ be the random variable representing the number of jacks drawn in $2$ trials with replacement.The total number of cards is $52$,and the num

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$X = x$$0$$1$$2$$P(X = x)$$\frac{144}{169}$$\frac{24}{169}$$\frac{1}{169}$

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(A) Let $X$ be the random variable representing the number of jacks drawn in $2$ trials with replacement.The total number of cards is $52$,and the number of jacks is $4$.The probability of drawing a jack in a single trial is $p = \frac{4}{52} = \frac{1}{13}$.The probability of not drawing a jack is $q = 1 - p = 1 - \frac{1}{13} = \frac{12}{13}$.Since the trials are independent (with replacement),$X$ follows a binomial distribution $B(n, p)$ where $n = 2$.$P(X = x) = ^nC_x p^x q^{n-x}$$P(X = 0) = ^2C_0 (\frac{1}{13})^0 (\frac{12}{13})^2 = 1 	imes 1 	imes \frac{144}{169} = \frac{144}{169}$$P(X = 1) = ^2C_1 (\frac{1}{13})^1 (\frac{12}{13})^1 = 2 	imes \frac{1}{13} 	imes \frac{12}{13} = \frac{24}{169}$$P(X = 2) = ^2C_2 (\frac{1}{13})^2 (\frac{12}{13})^0 = 1 	imes \frac{1}{169} 	imes 1 = \frac{1}{169}$Thus,the probability distribution is given by option $(A)$.

Two cards are drawn successively with replacement from a well-shuffled pack of $52$ cards. Then the probability distribution of the number of jacks is:

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$X = x$	$0$	$1$	$2$
$P(X = x)$	$\frac{144}{169}$	$\frac{24}{169}$	$\frac{1}{169}$