Win by two

Idea. From a tie, the next two points either decide the game (one player wins both) or bring back a tie. So the game is a repeated trial of pairs of points.

Step 1. With $p=0.4$ and $q=0.6$, the next two points are:

• both yours: probability $p^2=0.16$, you win;
• both your opponent's: probability $q^2=0.36$, you lose;
• split: probability $2pq=0.48$, tied again.

Step 2. Let $d$ be your probability of winning from a tie: $$d=p^2+2pqd,$$ so $$d=\frac{p^2}{1-2pq}=\frac{p^2}{p^2+q^2},$$ using $1=(p+q{)}^2=p^2+2pq+q^2$.

Step 3. $$d=\frac{0.16}{0.16+0.36}=\frac{16}{52}=\frac{4}{13}\approx \mathrm{0.308.}$$

Remark. Requiring two clear points amplifies the stronger player's edge: you win 40% of points but only about 31% of tied games. The same holds at deuce in tennis, which is reached from 30-30 as well.