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Assume that two tennis players and are struggling in a tournament and at some time, player has avantage on his service. The points are modelized by iid random variables taking values in , where if the point is won by and otherwise. Write and express the probability that wins the game in terms of . Application: if , would you bet that will win the game?

Recall that according to tennis rules, when players are at equality, the player wins the game if he can win two consecutive points from equality. If he only wins one point, we say that the advantage is given to .

If the score is advantage for , then, the event that the game is won by , corresponds to two distinct situations: either wins the next point and the game is won by or loses the next point, so that the score becomes equality but then, wins the game from equality. Mathematically, we obtain A$ wins from advantage to })=\alpha+(1-\alpha) \PP(\mbox{ wins from equality})$. It remains to calculate the probability that wins from equality. Now, the event that wins from equality can be split into two situations: either he scores two consecutive times or (he wins and loses) or (loses and wins) so that the score is again equality and from equality, finally wins the game. This can be mathematically written as AA$ wins from equality}) \PP(\mbox{ wins from equality})=\frac{\alpha^2}{1-2\alpha(1-\alpha)}$. Finally, AA$})=\alpha+(1-\alpha) \frac{\alpha^2}{1-2\alpha(1-\alpha)}=\frac{\alpha-\alpha^2(1-\alpha)}{1-2\alpha(1-\alpha)} $$