Question

Three cards are drawn successively, without replacement from a pack of 52 well-shuffled cards. What is the probability that the third card drawn is a queen given that first card is king of spades while second card is an ace?

• A. $\frac{2}{16575}$
• B. $\frac{3}{66300}$
• C. $\frac{1}{16575}$
• D. $\frac{2}{16575}$

Answer 1

The correct option is A $\frac{2}{16575}$

Let Q, K and A denote the event that the card drawn is a queen, king of spades and an ace respectively.
Probability of drawing king of spades is $P\left(K\right)=\frac{1}{52}.$

Now, total cards remaining in pack is 51.
Then probability of drawing an ace given that king of spades is already drawn is $P\left(A|K\right)=\frac{4}{51}.$

Now, total cards remaining in pack is 50.
Probability of drawing a queen given that a king of spades and an ace is already drawn is $P\left(Q|K\cap A\right)=\frac{4}{50}$

Now, by multiplication rule of probability,
$\begin{array}{ccc}P(Q\cap K\cap A)& =& P\left(K\right)\times P\left(A\right|K)\times P(Q|K\cap A)\\ & =& \frac{1}{52}\times \frac{4}{51}\times \frac{4}{50}\\ & =& \frac{2}{16575}\end{array}$