We roll a die until we get a $5$ and a $6$ for the first time, not necessarily consecutively and not necessarily in that order. We need to pay $x$ dollars before each die throw, and once both a $5$ and a $6$ have appeared for the first time, the game stops and we receive $100$ dollars. The problem is to determine $x$ such that this is a fair game.
My first thought has been to determine the expected number of rolls to get a $5$ and a $6$, which turns out to be $9$ (we need $3$ rolls on average to get a $5$ or a $6$ and then an additional $6$ to get both of them for the first time). However, I a having trouble linking this with a kind of recursive formula for the expected value of the game (obviously as a function of $x$), which would help us determine where $x$ should be set to make the game fair. Any ideas would be greatly appreciated.
First you are rolling until you hit a $5$ or $6$, which has probability $2/6$, so the average number of rolls is the inverse $6/2$.
Then you are rolling until you hit the other one, which has probability $1/6$, so the average number of rolls is the inverse $6/1$.
Hence, the total number of rolls on average is $9$. So in order for the game to be fair, you pay $100/9$ dollars per roll, since then on average people will pay $100$ dollars until the game ends, and their winnings will be zeroed out.
For a related problem, check out the coupon collector's problem.