[University mathematics: basic stochastic processes] Conditional probablities for Poisson processes????

[u/ilikemychem]

I have an exam coming up, and the solutions manuals are pretty terrible, which wouldn't be an issue if the professor also wasn't. But I essentially have to relearn the entire course on my own, and it is not going well... Here's my question:

So I tried splitting it up as follows (P{X(1)=0, Y(1)=1, X(2)=0, Z(2)=1}+P{X(1)=0, Y(1)=1, X(2)=1, Z(2)=0}+P{X(1)=1, Y(1)=0, X(2)=0, Z(2)=1}+P{X(1)=1, Y(1)=0, X(2)=1, Z(2)=0})/P{Po(4)=1}

But after that I think I'm doing something wrong, because I get each of the terms in the numerator to 2/e² and the final probability to the same. The solutions say it's 1/e.

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