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Assume that two tennis players $A$ and $B$ are struggling in a
tournament and at some time, player $A$ has avantage on his
service. The points are modelized by iid random variables $(X_i)$
taking values in $\{0,1\}$, where $X_i=1$ if the point is won by
$A$ and $X_i=0$ otherwise. Write $\alpha=\PP(X_i=1)$ and express the
probability that $A$ wins the game in terms of $\alpha$. Application:
if $\alpha=0.4$, would you bet that $A$ 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 $A$.

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

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