Here are a couple of curious related results about a generalized 2-player 1-set tennis game: the winner of the set is the first player to win $ n$ games, and the winner of each game is the first player to win $ k$ points. Let $ p>1/2$ be the probability that the stronger player wins a point, and let $ P(n,k,p)$ be the probability that the stronger player wins the set. Then
- $ P(n,k,p)>P(nk,1,p)=P(1,nk,p)$ if $ n,k>1$ , and
- $ P(n,k,p)>P(k,n,p)$ if $ n>k>1$ .
The way the paper goes about proving this is pretty clever, but I’m still wondering if there is a more direct combinatorial proof. Say, for a rational $ p$ , after clearing denominators, we are counting more of something on the left than on the right because of such-and-such an injection.
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