Sum

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

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#### 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; \[p = \frac{13}{52} = \frac{1}{4}; q = 1 - p = \frac{3}{4}\]

\[P(X = r) = ^{5}{}{C}_r \left( \frac{1}{4} \right)^r \left( \frac{3}{4} \right)^{5 - r} \]

\[ P(\text{ none is a spade } ) \hspace{0.167em} = P(X = 0)\]

\[ = ^{5}{}{C}_0 \left( \frac{1}{4} \right)^0 \left( \frac{3}{4} \right)^5 \]

\[ = \frac{243}{1024}\]

\[ = ^{5}{}{C}_0 \left( \frac{1}{4} \right)^0 \left( \frac{3}{4} \right)^5 \]

\[ = \frac{243}{1024}\]

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