Two players are playing tennis, and are at deuce. The first player has a 60% chance of winning each time they play a point, and the second players has a 40% chance. What is the probability that the first player wins the match?
I understand that there are two basic scenarios: Player 1 wins two consecutive points to cinch the set, OR {Player 1 Loses, Player 1 Wins (and ties it back to deuce), Player 1 Wins, Player 1 Wins}.
This sets up a sort of recursive logic, which I thought should be $p = 0.6^2 + 0.4(0.6)p$.
However, after attempting, I find that the solution has it as $p = 0.6^2 + 2(0.4)(0.6)p$, which I don't understand. Where does this 2 come from?
Best Answer
For player 1 to win the game, it can be three cases:
(1) win, win
(2) win, lose, (deuce again)
(3) lose, win, (deuce again)
$$p=0.6^2+0.6\cdot 0.4\cdot p+0.4\cdot 0.6\cdot p=0.6^2+2\cdot 0.6\cdot 0.4\cdot p$$