I'm trying to design a probability problem with two unknowns, one equality, and a unique solution. My first attempt:
Trillian has $n$ mice, of which $w$ are white. She chooses four at random. The probability that two are white is equal to the probability none are white.
As demonstrated elsewhere, the non-trivial solutions satisfy the polynomial
$$(2n-2w-5)^2-6(2w-1)^2=-5$$
This is Pell's equation, and it has an infinite number of integer solutions:
$$(n,w) \in \{ (8,2), (15,4), (53,15), \ldots \}$$
Alas. This problem doesn't work for my project.
But can we design a probability question with a similar feel (two unknowns and one equality) that has a unique solution?
