How to find $E[Y|X=1]$?

Question

A fair die is repeatedly rolled. Let $X$ and $Y$ denote, respectively, the number of rolls required to obtain a $1$ and a $2$. How do I find $E[Y|X=1]$?

edit: for using this

I got 1*6 = 6 total rolls.

Answer 1

You are asking for the expectation on the number of rolls needed to get a $2$ after the first roll got a $1$.

Obviously the first roll did not get a $2$, so you start over again with the second roll. The expected number of rolls after the first roll is one less than the expectation you want.

Define "got a $2$" as the success in a Bernoulli trial which has probability $\frac 16$ for each trial. You should know the formula for the expected number of trials before the first success for Bernoulli trials. Calculate that, add one, and there is your answer.

Let us know if you do not actually know the formula for the expected number of trials before the first success in a series of Bernoulli trials.