Question

Five cards are drawn successively with replacement from a well-shuffled pack of 52 cards. What is the probability that  only 3 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; \[p = \frac{13}{52} = \frac{1}{4}; q = 1 - p = \frac{3}{4}\]

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

\[ P(\text{ only 3 cards are spades)}  = P(X = 3) \]
\[ =^{5}{}{C}_3 \left( \frac{1}{4} \right)^3 \left( \frac{3}{4} \right)^2 \]
\[ = \frac{1}{1024}\left( 90 \right) \]
\[ = \frac{45}{512}\]