You have a 30% chance of winning a raffle. If you buy 3 tickets, what is the probability you win at least once?

Example Answer

I would first ask whether the three tickets are independent chances or entries in the same draw. If each is an independent trial with a 30% chance of winning, the chance of no wins is 0.7 cubed. Taking the complement gives 1 minus 0.343, or 65.7%, for at least one win.

For tickets in one raffle, that calculation may be wrong. With one winning ticket among N equally likely tickets, owning three distinct tickets gives a 3/N chance of winning. For example, three of ten tickets gives 30%. If there are several winning tickets, I need the total ticket count and draw rules. A 30% marginal probability by itself does not tell me how the three outcomes depend on each other.

Why this works

It gives the familiar complement calculation while making clear when it applies. The raffle rules determine the joint probabilities.

Follow-up to prepare for

With W winning tickets among N tickets, the chance of at least one winner in three tickets is 1 − C(N−W, 3)/C(N, 3), when three distinct tickets are sampled without replacement.