Exactly two red cards in three

Question

A small deck has 15 cards: 5 are red and 10 are black. You deal 3 cards at random. What is the probability that exactly 2 of the 3 cards are red?

Accepted Answer

Dealing 3 cards is an unordered pick, so count groups with combinations, where \binom{n}{k} counts ways to choose k from n. Total ways to choose 3 cards from 15: \binom{15}{3}=455. Favorable: exactly 2 red and 1 black. Pick 2 of the 5 red and 1 of the 10 black: \binom{5}{2}	imes\binom{10}{1}=10	imes 10=100. Probability: \frac{100}{455}=\frac{20}{91}\approx 0.220. Takeaway. The same split works for any target count: choose the reds you want from the red pile, fill the rest from the black pile, and divide by all ways to deal the hand.

Exactly two red cards in three

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