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 ?

Answer 1

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

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

\[ P(\text{ All cards are spades } ) \hspace{0.167em} = P(X = 5) \]
\[ = ^{5}{}{C}_5 \left( \frac{1}{4} \right)^5 \left( \frac{3}{4} \right)^0 \]
\[ = \frac{1}{1024}\]