A Long Chess Match
Problem 661
Two friends and are great fans of Chess. They both enjoy playing the game, but after each game the player who lost the game would like to continue (to get back at his opponent) and the player who won would prefer to stop (to finish on a high).
So they come up with a plan. After every game, they would toss a (biased) coin with probability of Heads (and hence probability of Tails). If they get Tails, they will continue with the next game. Otherwise they end the match. Also, after every game the players make a note of who is leading in the match.
Let denote the probability of winning a game and the probability of winning a game. Accordingly is the probability that a game ends in a draw. Let denote the expected number of times was leading in the match.
For example, and , both rounded to six places after the decimal point.
For example, and , both rounded to six places after the decimal point.
Let
For example , rounded to 4 digits after the decimal point.
For example , rounded to 4 digits after the decimal point.
Find , rounded to 4 digits after the decimal point.