Three cards are drawn successively, without replacement from a pack of 52 well shuffled cards. What is the probability that first two cards are kings and the third card drawn is an ace?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → $\frac{2}{5525}$ ##
Let $K$ denote the event that the card drawn is king and $A$ be the event that the card drawn is an ace. Clearly, we have to find $P(KKA)$.
Now $P(K) = \frac{4}{52}$
Also, $P(K|K)$ is the probability of second king with the condition that one king has already been drawn. Now there are three kings in $(52 - 1) = 51$ cards.
Therefore $P(K|K) = \frac{3}{51}$
Lastly, $P(A|KK)$ is the probability of third drawn card to be an ace, with the condition that two kings have already been drawn. Now there are four aces in left 50 cards.
Therefore $P(A|KK) = \frac{4}{50}$
By multiplication law of probability, we have
$P(KKA) = P(K) \cdot P(K|K) \cdot P(A|KK)$
$= \frac{4}{52} \times \frac{3}{51} \times \frac{4}{50} = \frac{2}{5525}$