  The birthday paradox states that there is at least a 50% chance that some two
out of twenty-three randomly chosen people will share the same birth date. The
calculation for this problem assumes that all birth dates are equally likely.
We consider the following two modifications of this question. If the
distribution of birthdays is non-uniform, does that increase or decrease the
probability of matching birth dates? Further, what if we focus on birthdays
shared by some particular pairs rather than any two people. Does a non-uniform
distribution on birth dates increase or decrease the probability of a matching
pair? In this paper we present our results in this generalized setting. We use
some results and methods due to Sokal concerning bounds on the roots of
chromatic polynomials to prove our results.
