If a player is 50% as good as I am at a game, how many games will it be before she finally wins one game? This is a real life problem. I play Foosball with my colleague who hasn't beaten me so far. I have won 18 in a row. She is about 50% as good as I am (the average margin of victory is 10-5 for me). Mathematically speaking, how many games should it take before she finally wins a game for the first time?

Here's one plausible model: each new point (independent of past history) has probability $2/3$ of being scored by you and $1/3$ of being scored by her.  The game ends, I assume, when one player has $10$ points.  The probability that she wins by a score of $10$ to $x$, i.e. that of the first $9+x$ points she scores exactly $9$ and then she wins the last point, is
${{9+x} \choose 9} (1/3)^{10} (2/3)^x$.  The total probability that she wins the game is
then $\displaystyle p = \sum_{x=0}^9 {{9+x} \choose 9} 2^x/3^{10+x} = \frac{75275227}{1162261467} \approx 0.0648$.  The expected number of games until she wins is $1/p \approx 15.44$.
