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(A) Let $X$ represent the number of spade cards among the five cards drawn. Since the drawing of cards is with replacement,the trials are Bernoulli tr

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$\frac{1}{1024}$

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(A) Let $X$ represent the number of spade cards among the five cards drawn. Since the drawing of cards is with replacement,the trials are Bernoulli trials.In a well-shuffled deck of $52$ cards,there are $13$ spade cards.Therefore,the probability of drawing a spade in a single trial is $p = \frac{13}{52} = \frac{1}{4}$.The probability of not drawing a spade is $q = 1 - p = 1 - \frac{1}{4} = \frac{3}{4}$.$X$ follows a binomial distribution with $n = 5$ and $p = \frac{1}{4}$.The probability mass function is given by $P(X = x) = ^{n}C_{x} q^{n-x} p^{x}$,where $x = 0, 1, 2, 3, 4, 5$.We want to find the probability that all five cards are spades,which is $P(X = 5)$.$P(X = 5) = ^{5}C_{5} \left(\frac{3}{4}\right)^{5-5} \left(\frac{1}{4}\right)^{5}$.$P(X = 5) = 1 \cdot \left(\frac{3}{4}\right)^{0} \cdot \left(\frac{1}{4}\right)^{5}$.$P(X = 5) = 1 \cdot 1 \cdot \frac{1}{1024} = \frac{1}{1024}$.

Five cards are drawn successively with replacement from a well-shuffled deck of $52$ cards. What is the probability that all the five cards are spades?

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