# What is the expected number of d20 rolls needed for at least one 1 and at least one 20 to come up?

Let's say you're playing D&D. You know that for any fair n-sided die, it will take an average of n times before you roll a given face on that die (because it's a geometric distribution). So if you wanted to roll a natural 20, across all the sessions you play it will take an average of 20 rolls of a 20-sided die (a d20) per session before you see a natural 20 for the first time. Each individual session may take more or less rolls to get a natural 20 (let's assume you're rolling a LOT per session, like at least 100 times, and your campaign has infinite sessions), but on average it will take you 20 rolls for the number 20 to show up for the first time in a given session.

Now let's say you want to predict, on average, when your first natural 20 AND your first natural 1 will show up. In other words, what is the expected number of rolls of a d20 needed for at least one 1 AND at least one 20 to come up?

They don't need to appear consecutively; they can appear in any order. You're also only rolling one d20, one roll at a time. Any face can come up any number of times, but you only stop rolling once you've seen 1 AND 20 at least once each.

Consider the event that a $$1$$ or a $$20$$ shows up. It has probability $${1\over 10}$$, so the expected number of rolls until the first of the two numbers comes up is $$10$$. Then the expected number of rolls until the other numbers shows up is $$20$$, as you said. In all, the expected number of rolls is $$30.$$