As a condition for the acceptance to a tennis club a novice player N is set to meet two members of the club, G (good) and T (top, i.e. better than good) within a total of three games (i.e. at most three!).
In order to be accepted, N must win against both G and T in two successive games.
N is free to choose with whom to start: T or G.
Which one is preferable?

Attributed to the late Leo Moser (1921—1970)

Let N's probability of winning any given game against G be g.
Let N's probability of winning any given game against T be t.

g>t because G is not as good a player as T.

If the sequence of games is GTG, then N must win the game against T and either one of the two games against G. This probability is then t*(2*g-g^2). The subtraction of g^2 is to subtract out the double counting of winning both of these games.

Likewise, if the sequence of games is TGT, then N must win the game against G and either one of the two games against T. This probability is then g*(2*t-t^2).

The GTG probability comes out to 2*g*t - t*g^2, while the TGT probability comes out to 2*g*t - g*t^2.  The first term in each of these is the same. The second, negative, term is larger when the g factor is squared than when the t factor is squared, so more is subtracted from the probability in the GTG case. So the probability of N winning is greater in the TGT case, and N should choose to play against T first.

Posted by Charlie on 2010-09-08 00:15:11