wu :: forums (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi) riddles >> putnam exam (pure math) >> Disjoint Sets (Message started by: Barukh on Dec 22nd, 2013, 12:19am)  Title: Disjoint Sets Post by Barukh on Dec 22nd, 2013, 12:19am Consider a set of N different values.By randomly choosing elements from N, two subsets – A and B – are formed, so that |A|*|B| = N, and |A|, |B| > Na for some constant a (e.g. their sizes depend on N).What is the probability that A and B are disjoint, when N is big? Title: Re: Disjoint Sets Post by Michael Dagg on Dec 26th, 2013, 8:58am Comments... Title: Re: Disjoint Sets Post by Michael Dagg on Dec 26th, 2013, 10:43am I've thought of another way to see that the limitingprobability of 1/e is reasonable.  Choose a random set A of legal size (i.e. of size N^\alpha, where a < \alpha < 1-a).  Then begin constructing B by choosing random elements from S.  At each choice, the probability of choosing a member of A is N^(\alpha - 1). If we keep track of the number of choices X that are members of A, then X is Poisson distributed with parameters n = N^(1-\alpha) and p = N^(\alpha - 1).  Therefore, the probability of choosing no member of A is e^(-np) = 1/e. The only reason this isn't exact is that when constructing B, we might choose an element more than once.   One can show that the expected number of elements of S that are chosen more than once is quite small, so this shouldn't affect the limiting probability. Powered by YaBB 1 Gold - SP 1.4! Forum software copyright © 2000-2004 Yet another Bulletin Board