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)

The probabilities are the same either way.

Let T = probability of beating Player T
Let G = probability of beating Player G

There are three outcomes for each player:

Lose, Win, Win    |    Win, Lose, Win     |   Win, Win, ---

For playing Player T first:

[(1-T) * (T) * (G)] + [(T) * (1-G) * (G)] + [(T) * (G)] = TG * (3 - T - G)

For playing Player G first:

[(1-G) * (G) * (T)] + [(G) * (1-T) * (T)] + [(G) * (T)] = TG * (3 - T - G)

The probabilities are the same.

Posted by hoodat on 2010-09-07 12:12:55