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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

Vedclass · Accepted Answer

$\frac{45}{512}$

Vedclass · Answer

(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.Thus,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 $q = 1 - p = 1 - \frac{1}{4} = \frac{3}{4}$.Here,$n = 5$ and $X$ follows a binomial distribution $B(n, p)$.The probability mass function is given by $P(X = x) = ^{n}C_{x} \cdot q^{n-x} \cdot p^{x}$.We need to find the probability that exactly $3$ cards are spades,i.e.,$P(X = 3)$.$P(X = 3) = ^{5}C_{3} \cdot (\frac{3}{4})^{5-3} \cdot (\frac{1}{4})^{3}$.$P(X = 3) = 10 \cdot (\frac{3}{4})^{2} \cdot (\frac{1}{4})^{3}$.$P(X = 3) = 10 \cdot \frac{9}{16} \cdot \frac{1}{64}$.$P(X = 3) = 10 \cdot \frac{9}{1024} = \frac{90}{1024} = \frac{45}{512}$.

Five cards are drawn successively with replacement from a well-shuffled deck of $52$ cards. What is the probability that only $3$ cards are spades?

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