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 Tennis,anyone? (Posted on 2010-09-07)
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)

 Submitted by Ady TZIDON Rating: 5.0000 (1 votes) Solution: (Hide) Charlie provided fully explained solution: 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.

 Subject Author Date solution Charlie 2010-09-08 00:15:11 re(2): Not so fast ed bottemiller 2010-09-07 21:51:28 Whose serve? ed bottemiller 2010-09-07 16:40:22 re: Not so fast hoodat 2010-09-07 13:05:14 Not so fast Steve Herman 2010-09-07 12:47:18 Solution hoodat 2010-09-07 12:12:55 solution Daniel 2010-09-07 11:47:01

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