Two cards are drawn at random, one after the other, without replacement from a well-shuffled pack of 52 playing cards. The probability that the first card is a king and the second card is a queen is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{4}{663}\)
Let K be the event that the first card is a king and Q the event that the second card is a queen. By the multiplication theorem, \(P(K\cap Q) = P(K)\,P(Q|K)\).
\( \displaystyle P(K) = \frac{4}{52} \)
After a king is removed, 51 cards are left and all 4 queens are still there:
\( \displaystyle P(Q|K) = \frac{4}{51} \)
So
\( \displaystyle P(K\cap Q) = \frac{4}{52}\times\frac{4}{51} = \frac{1}{13}\times\frac{4}{51} = \frac{4}{663} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{1}{169}\) — Incorrect. This is \(\frac{4}{52}\times\frac{4}{52}\), the answer with replacement.
Option (3) \(\displaystyle\frac{8}{663}\) — Incorrect. This is the probability of a king and a queen in either order; the question fixes the order.
Option (4) \(\displaystyle\frac{1}{221}\) — Incorrect. This is \(\frac{4}{52}\times\frac{3}{51}\), the probability that both cards are kings.