Question

Two cards are drawn at random from an ordinary deck of 52 playing cards. What is the probability of getting two aces if

(a) the first card is replaced before the second card is drawn;
(b) the first card is not replaced before the second card is drawn?

Answer 1

(a) Since there are four aces among the 52 cards, we get
\[
\frac{4}{52} \cdot \frac{4}{52}=\frac{1}{169}
\]

(b) Since there are only three aces among the 51 cards that remain after one ace has been removed from the deck, we get
\[
\frac{4}{52} \cdot \frac{3}{51}=\frac{1}{221}
\]
Note that
\[
\frac{1}{221} \neq \frac{4}{52} \cdot \frac{4}{52}
\]
so independence is violated when the sampling is without replacement.