From a pack of 52 playing cards, a card is drawn at random. What is the…
20172021
From a pack of 52 playing cards, a card is drawn at random. What is the probability that the card is a spade or an ace?
- A.
4/13
- B.
6/13
- C.
8/13
- D.
2/5
Attempted by 4 students.
Show answer & explanation
Correct answer: A
The addition rule (inclusion-exclusion) for two events states P(A or B) = P(A) + P(B) - P(A and B): when two events can overlap, simply adding their individual probabilities double-counts the outcomes common to both, so that overlap must be subtracted exactly once.
Let A be the event "the card is a spade" and B be the event "the card is an ace". A standard deck has 52 cards.
Number of spades = 13, so P(A) = 13/52.
Number of aces = 4, so P(B) = 4/52.
The only card that is both a spade and an ace is the Ace of Spades, so P(A and B) = 1/52.
By the addition rule, P(A or B) = 13/52 + 4/52 - 1/52 = 16/52 = 4/13.
Direct counting confirms this: the 13 spades already include the Ace of Spades, and the only additional aces are the Ace of Hearts, Ace of Diamonds, and Ace of Clubs (3 more cards). That gives 13 + 3 = 16 favourable cards out of 52, i.e. 4/13 - the same result.