### 538 Riddler: Defeating Roger Federer

#### 07 July ’16

I began this challenge at the obvious starting point: I win my first 71 points against Roger. This puts me up 2 sets to 0, up 5-Nill in the 3rd, and winning 40-Love in what is possibly the final game of the match. Things look pretty good; I've got a triple match point. However, Roger's greatness cannot be underestimated, even in his twilight years. The probability that we win this specific game is:

This technically represents the lower bound of our win probability. However, the probability that I prevail if I lose this game is infinitesimally small, so this reprsents a good approximation.

However, if we focus exclusively on that final game, there is a scenario which gives us a few more opportunities to win. We can begin the match up 2 sets to 0 and up 6-0 in the tie break in the 3rd (tie breaks are allowed in all sets except the 5th at Wimbledon). This gives us 6 match points! The probability that we win this tie break is:

When the probability of winning each point hits 11%, we are more likely to prevail in the tie break than Roger! Cheers.