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Probability Question

Posted By: Timothy Chow
Date: Tuesday, 29 November 2011, at 3:24 p.m.

In Response To: Probability Question (Ray Kershaw)

Ray asked:

Is there a simple answer to David's question: given a draw of 64, what is the probability that four specific players are each in a different quarter of the draw ?

The answer to this question might not be obvious when you ask it out of the blue, but most of the work has already been done in this thread.

Call the above probability Q (for "quarter"). Call the original probability that David asked about (and that Casper and Sam calculated) S (for "semifinal"). Finally, let P be the probability that your favorite four players reach the semifinals, given that they are in different quarters of the draw. Then it is easy to see that P = 1/164, and also that S = P*Q. We can then solve to get Q = S/P = 164/(64 choose 4).

Alternatively, if you look carefully at Casper's calculation, you'll see that the first four terms of his product show you how to calculate it.

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