Question

Suppose you have a 1/10 chance of winning with a scratch-off lottery ticket. If you buy 3 tickets, what is the probability of winning with all 3?

Answer 1

There are a couple of ways of working this out:
1. Drawing a probability tree diagram
2. Using a quick shortcut

If we were to draw the tree, the first two branches would have a 1/10 on one branch (meaning a win) and a 9/10 on the other branch (meaning no win) and these two branches would come off each branch respectively, and again off those branches. If you can't visualise what I just said (fair enough!) then you can look up some images on Google or a textbook.

Let's use the shortcut though. The probability of winning with ANY ticket is the same as the OPPOSITE probability (or the complementary event) of losing with all 3. This is the same with many probability scenarios: P(either) = 1 - P(none).

So, let's take a more simple question - what's the probability of scratching a ticket, not winning, scratching the next ticket, still not winning, and scratching the last ticket, not winning again? You just multiply the probabilities together. Our final answer would be 1 - (no wins).