2 player game

Suppose two players play a game alternating turns. Player 1 goes first and has 50% chance of winning in that round. Player 2 goes second (if P1 doesn’t win) and has 33% chance of winning. They keep alternating in this way until one player wins. What is the probability that P1 wins?

Solution

This is a very standard question, and so we can use a standard tool: setting up and solving the recursive equation.

\[\begin{gathered} \mathrm{Pr}(\text{P1 wins}) = p_1 + (1-p_1)(1-p_2)\mathrm{Pr}(\text{P1 wins}) \end{gathered}\]

which gives the answer $p_1/(p_1+p_2-p_1p_2)$, in this case $3/4$.

Another variation is that there are not alternating turns. Instead, for each turn of the game, Player 1 has probability $p_1$ of winning, Player 2 has probability $p_2$ pf winning, otherwise they play another turn.