At least one winning raffle ticket

Question

A bin has 12 raffle tickets, of which 4 are winners and 8 are duds. You buy 3 tickets at random. What is the probability that at least one of your 3 tickets is a winner?

Accepted Answer

Rather than adding up "exactly 1 winner," "exactly 2," and "exactly 3," find the chance of "no winners" and subtract from 1. Chance of zero winners means all 3 tickets come from the 8 duds. With combinations, where \binom{n}{k} counts ways to choose k from n: P(	ext{no winner})=\frac{\binom{8}{3}}{\binom{12}{3}}=\frac{56}{220}=\frac{14}{55}. Subtract from 1: 1-\frac{14}{55}=\frac{41}{55}\approx 0.745. Takeaway. Even with four winners in the bin, buying three tickets gives you roughly a three-in-four shot at catching at least one. The complement makes the arithmetic short.

At least one winning raffle ticket

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