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Topic: Hypergeometric? No! But how close?
Replies: 0

 Bj|rn Elstad Posts: 2 Registered: 12/12/04
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