At least one heart in two cards

Question

You draw two cards from a well-shuffled standard 52-card deck. You are told, truthfully, that at least one of them is a heart. Given that, what is the probability that both cards are hearts?

Accepted Answer

Conditional probability is just favorable outcomes divided by the outcomes still allowed by the clue: P(	ext{both hearts} \mid 	ext{at least one heart}) = \frac{P(	ext{both hearts})}{P(	ext{at least one heart})}. Count unordered two-card hands. There are \binom{52}{2} = 1326 hands in all. Both hearts. Choose 2 of the 13 hearts: \binom{13}{2} = 78 hands. At least one heart. Easier by its complement (no hearts): \binom{39}{2} = 741 heart-free hands, so 1326 - 741 = 585 hands contain at least one heart. Divide: \frac{78}{585} = \frac{2}{15} \approx 0.133. So even after learning a heart is pr…

At least one heart in two cards

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