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Hypergeometric? No! But how close?
Posted:
Jun 18, 1996 9:01 AM


"Bingo" lottery. Players chose a=20 numbers from N=75 numbers, after some rules that are not of consern here. Approx 20% of the 8 10^17 { N \choose a } subsets are legal. There are many players, 10^6. Numbers are drawn from the N. A "winner" is one player that has all his a=20 numbers among the drawn ones.
One can consider the hypergeometric probability for x "winners". hyp(x; s, m, t) = {{m \choose x } { s  m \choose t  x}} \over { s \choose t} where s = { N \choose a}, m the players, t = {n \choose a}.
In a correct drawing process of hypergeometric sample, every element, i.e {n \choose a }, in the sample is drawn "random" (with removing). In the drawing process above one draws n numbers, and consider the big set of all {n \choose a } subsets. The "randomness" is limited, compared with a correct hypergeometric sample.
The problem: When is hyp(x; s, m, t) a good approximation to the probability of x winners? How good? (It is very good when n=N)!
 bjoern



