First of all, to give credit, problem taken from this app:
https://play.google.com/store/apps/details?id=atorch.statspuzzles&hl=en_GB(It's puzzle 10 from section 2).
Anyhow, to cut to the chase. Given 0 < p < 1, let X~Binom(4, p), Y~Binom(6, p), X and Y independent.
The problem is to find for what value of p does P(X>=2) = P(Y>=3)?
I found the solution by forming the obvious expressions for the two probabilities, and then a fairly painful algebra grind to factorize. But it was NOT nice, and without using Wolfram-Alpha to guide things I'm not sure I'd have managed it at all.
But, it does turn out that the actual factorization is quite nice:
It feels there should be a way of finding this without the algebra grind, but I'm not really seeing it.