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Question
Five cards are drawn successively with replacement from a well-shuffled pack of 52 cards. What is the probability that all the five cards are spades ?
Solution
Let X denote the number of spade cards when 5 cards are drawn with replacement. Because it is with replacement,
X follows a binomial distribution with n = 5; \[p = \frac{13}{52} = \frac{1}{4}; q = 1 - p = \frac{3}{4}\]
\[ = ^{5}{}{C}_5 \left( \frac{1}{4} \right)^5 \left( \frac{3}{4} \right)^0 \]
\[ = \frac{1}{1024}\]
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