Author: Ratko V Tomic
Date: 15:01:48 04/26/02
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The error sqrt(N) is valid in the limit of large N (where Stirling approximation for N! is usable). In that case edge effects don't occur provided 1/sqrt(N)<<MIN(y,(1-y)). For small N you have to use the exact combinatorics to get probabilities of different outcomes of the sequence of n rounds (e.g. the binomial distribution: P(k)=Comb(N,k)*y^k*(1-y)^(N-k)) for k wins, where Comb(N,k)=N!/[(N-k)!*k!]). The most probable value for k is kmax=y*N. The next most probable values are kmax+/-1, etc. Adding these probabilities around kmax (in order of decreasing P(k)) until the sum reaches say 0.95 will give you the most accurate error margins for any desired confidence interval and any N. A small C or Basic program can compute this quickly for any practical match length.
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