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

Title: Re: Disjoint Sets
Post by Michael Dagg on Dec 26th, 2013, 10:43am
I've thought of another way to see that the limiting
probability 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.