# Dice Game: Probability of rolling three 6's before my opponent

Let's say I'm playing a dice game with a friend. We each get two dice, and we roll all of our dice at the same time. You get one point for each 6 you have. First to score at least three points wins. If both of us get to 3 (or more) points on the same trial, it is a tie.

1. What is the probability that I win? lose? tie?
2. What is the expected number of rolls required for either of us to win?
3. If I replace one of my dice with an unfair dice which rolls 6 2/6 times, how does that affect the above answers?

More info: I'm actually trying to design a slot game (5 reels). If a particular symbol appears on reels 1 or 2, the player gets a point, if the symbol appears on reels 4 or 5 the computer gets a point. First to three points wins. In order to balance the payout of the machine, I need to know the number of trials expected before the game ends, and how often the player wins vs the computer wins or ties.

• For 1. by symmetry you know that the probability that you lose is the same as the probability that you win. What's left to figure out is the probability that you tie. If $p_t$ is the probability of a tie, then the probability that you win is $\frac{1}{2}(1-p_t)$. Jun 22, 2015 at 19:56