Guidance for the probability of winning dice game?

Disclaimer: I am not looking for a solution to this, just a clue to help me arrive by myself to the right solution.

Two players are rolling a die, and the first one to get three ones, wins. So far, player A has gotten two ones, and player B has gotten one. What is the probability that player A wins?

What I am struggling with is the fact that the game can go on forever, so I am not sure of how to approach the problem. Any guidance is greatly appreciated.

• Have you heard of a geometric random variable? – Indefinite Sep 10 '17 at 12:20
• "The game will go on forever" should translate to an infinite geometric series in your solution. – iamwhoiam Sep 10 '17 at 12:21
• I have not heard of it @Indefinite , and through the research I've been doing, it seems a geometric series is not strictly necessary? – Bee Sep 10 '17 at 12:22
• Letting $P_A(a,b)$ (resp. $P_B(a,b)$) denote the probability that $A$ wins given that $A$ has thrown $a$ ones and $B$ has thrown $b$ given that it is $A's$ turn (resp. $B's$ turn), we get a simple backwards induction. – lulu Sep 10 '17 at 12:23
• Note: you did not specify whose turn it was. That is clearly relevant. – lulu Sep 10 '17 at 12:26