X is non-negative with mgf = (1/1-2t)e^(t/1-2t) t<1/2

Pr(Y=y|X=x) = Poisson with parameter 2X

Find Pr(Y=0)

Any hints on where to start? I applied law of total probability etc, but can't get Pr(X=x) from the mgf, or can I?

Don't necessarily want to have this exact problem solved as I want to be able to solve it myself, however looking for insight into where to start the thinking process from in order to solve similar problems.

Thanks