At least one counterfeit coin in two

Question

A pile has 9 coins, of which 3 are counterfeit and 6 are genuine. You draw 2 coins at random. What is the probability that at least one of the 2 coins is counterfeit?

Accepted Answer

Find the chance of the opposite event, "both coins genuine," and subtract from 1. Chance of zero counterfeits means both coins come from the 6 genuine ones. With combinations, where \binom{n}{k} counts ways to choose k from n: P(	ext{both genuine})=\frac{\binom{6}{2}}{\binom{9}{2}}=\frac{15}{36}=\frac{5}{12}. Subtract from 1: 1-\frac{5}{12}=\frac{7}{12}\approx 0.583. Takeaway. Even a small draw of two from a pile that is one-third fake gives better-than-even odds of catching a counterfeit. Counting "none" first is the shortcut.

At least one counterfeit coin in two

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