Probability of $A$ winning My department is not math so I can not use terminology correctly. I have a question. I hope that you can help me. 

There are two players. In the game, probability of winning an experiment is proportional to the points of a player. At the end of each round the loser will give one point to the winner and the game ends when any player is $0$.
  At first $a =1$ and $b=2$; What is the chance of winning?

Thanks in advance.

Let $p$ be the probability of A winning.  For this to happen, A must win the first round, probability $\frac13$.  Then the situation is A $2$ points, B $1$ point.  This is the same as at the start, except with players reversed.  So at this stage the probability of B winning is $p$ and the probability of A winning is $1-p$.  So
$$p=\frac13(1-p)\ ,$$
and solving gives $p=\frac14$.
