Question

Two cards are drawn at random from a pack of 52 playing cards. If both the cards drawn are found to be black in colour, then the probability that at least one of them is a face card is:

• A. \(\frac{3}{13}\)
• B. \(\frac{3}{5}\)
• C. \(\frac{9}{65}\)
• D. \(\frac{27}{65}\)

Hint:

Calculate complement probability for "at least one" scenarios to simplify the problem.

Answer 1

The Correct Option is D

Step 1: Identify black cards and face cards
Number of black cards = 26 (clubs and spades).
Number of black face cards = 3 face cards each in clubs and spades = 6. Step 2: Total ways to choose 2 black cards
\[ \binom{26}{2} = \frac{26 \times 25}{2} = 325 \] Step 3: Ways to choose 2 black cards with no face cards
Number of black non-face cards = \(26 - 6 = 20\).
\[ \binom{20}{2} = \frac{20 \times 19}{2} = 190 \] Step 4: Probability that at least one is face card
\[ = 1 - \text{Probability(no face cards)} = 1 - \frac{190}{325} = \frac{135}{325} = \frac{27}{65} \]