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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

$\frac{243}{1024}$

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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.Probability of drawing a spade,$p = \frac{13}{52} = \frac{1}{4}$.Probability of not drawing a spade,$q = 1 - p = 1 - \frac{1}{4} = \frac{3}{4}$.$X$ 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 none is a spade,which is $P(X = 0)$.$P(X = 0) = ^{5}C_{0} \cdot (\frac{3}{4})^{5-0} \cdot (\frac{1}{4})^{0}$.$P(X = 0) = 1 \cdot (\frac{3}{4})^{5} \cdot 1 = \frac{243}{1024}$.Thus,the probability that none of the cards is a spade is $\frac{243}{1024}$.

Five cards are drawn successively with replacement from a well-shuffled deck of $52$ cards. What is the probability that none is a spade?

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