A card is drawn at random from a well-shuffled pack of 52 playing cards. Consider the following statements:
(A) The probability that the card is a king, given that it is a face card, is \(\displaystyle\frac{1}{3}\).
(B) The probability that the card is a spade, given that it is a black card, is \(\displaystyle\frac{1}{4}\).
(C) The probability that the card is a face card, given that it is a heart, is \(\displaystyle\frac{3}{13}\).
(D) The probability that the card is an ace, given that it is a spade, is \(\displaystyle\frac{1}{4}\).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (C) only
In each statement the given condition shrinks the sample space to the cards of that kind, and the probability is the number of favourable cards among them divided by their number.
Face cards: 12 (J, Q, K of 4 suits), of which 4 are kings
Black cards: 26, of which 13 are spades
Hearts: 13, of which 3 are face cards
Spades: 13, of which 1 is an ace
So the four conditional probabilities are \(\frac{4}{12} = \frac{1}{3}\), \(\frac{13}{26} = \frac{1}{2}\), \(\frac{3}{13}\) and \(\frac{1}{13}\).
Checking the statements
(A) The probability that the card is a king, given that it is a face card, is \(\frac{1}{3}\). — Correct. 4 kings among 12 face cards give \(\frac{4}{12} = \frac{1}{3}\).
(B) The probability that the card is a spade, given that it is a black card, is \(\frac{1}{4}\). — Incorrect. 13 spades among 26 black cards give \(\frac{1}{2}\).
(C) The probability that the card is a face card, given that it is a heart, is \(\frac{3}{13}\). — Correct. 3 face cards (J, Q, K) among 13 hearts give \(\frac{3}{13}\).
(D) The probability that the card is an ace, given that it is a spade, is \(\frac{1}{4}\). — Incorrect. 1 ace among 13 spades gives \(\frac{1}{13}\).