Probability of drawing a face card or a spade

Question

You draw a single card from a well-shuffled standard 52-card deck. Face cards are Jack, Queen, and King. What is the probability that it is a face card or a spade?

Accepted Answer

There are 12 face cards (3 in each of the 4 suits) and 13 spades. Simply adding 	frac{12}{52} + 	frac{13}{52} double-counts the cards that are both a face card and a spade: the Jack, Queen, and King of spades, three cards. Inclusion-exclusion corrects the overlap: P(F \cup S) = P(F) + P(S) - P(F \cap S) = \frac{12}{52} + \frac{13}{52} - \frac{3}{52} = \frac{22}{52} = \frac{11}{26} \approx 0.423. Counting directly agrees: the 13 spades, plus the 9 non-spade face cards, give 22 favorable cards out of 52. Takeaway. P(A \cup B) = P(A) + P(B) - P(A \cap B). Whenever two categories can overlap (h…

Probability of drawing a face card or a spade

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