Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.2 9/12/84; site aero.ARPA
Path: utzoo!watmath!clyde!burl!ulysses!unc!mcnc!decvax!decwrl!Glacier!oliveb!hplabs!sdcrdcf!trwrb!aero!sinclair
From: sinclair@aero.ARPA (William S. Sinclair)
Newsgroups: net.math
Subject: Probability Problem; Reward!
Message-ID: <523@aero.ARPA>
Date: Wed, 30-Oct-85 19:38:43 EST
Article-I.D.: aero.523
Posted: Wed Oct 30 19:38:43 1985
Date-Received: Sun, 3-Nov-85 08:57:35 EST
Reply-To: sinclair@aero.UUCP (William S. Sinclair)
Organization: The Aerospace Corp., El Segundo, CA
Lines: 19

I will send $20 to the first person who gives an analytic solution to this
problem:

Suppose we pick n points from a uniform distribution in two dimensions, for
example (0,1) in both x and y. Now form the convex hull of these. k of these
points will be inside the convex hull, while n-k will be ON the hull.

What is the formula for P(n,k), the probability that exactly k points out of
n will be inside?

Hint (1): p(3,3)=1.0
Hint (2): As n gets large, the expected value of k grows as sqrt(n).

I will check your formula by using a Monte Carlo simulation.

Bill S.
Go ahead and flame! I brought marshmallows!!
213/647-1753