Solution of an odds inversion problem

Consider the problem of determining the possible numbers of balls of two different colors in an urn such that if two are drawn out at random, the odds that they are different colors are a given value. We present a general solution of this problem for all odds from nil to certainty. The solution methods use relatively simple concepts from number theory such as modular inverses and the Pell equation. We find upper bounds on the number of solutions and the magnitude of solutions for those cases that have at most a finite number of solutions. We also define solution classes for cases that have an infinite number of solutions, and identify cases having a determinate number of solution classes.