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)

perhaps my reasoning is wrong, but according to my determinations it does not matter which he chooses to play first.

My reasoning is as follows:
Let Gw,Gl represent a win/loss against G and
Tw,Tl a win/loss against T

Now if he plays G first then the possible paths to acceptance are
Gw,Tw
Gw,Tl,Tw
Gl,Gw,Tw

now if instead he plays T first then the paths to acceptance are
Tw,Gw
Tw,Gl,Gw
Tl,Tw,Gw

now these are just permutations of the 3 paths for playing G first, thus when you compute the probabilities for each path they are the same.  Thus the overall odds of acceptance are the same regardless of who he plays first.
if G is the odds of winning against G and T against T then the overall odds of accceptance are
GT+G(1-T)T+(1-G)GT

Posted by Daniel on 2010-09-07 11:47:01