$X$ and $Y$ are two independent Poisson Processes with rates $\lambda_1, \lambda_2$ respectively. What is the probability that at least two events from $X$ occur before a total of two events from $Y$? A valid example would be $X_1, Y_1, X_2, Y_2...$ since two events from $X$ occurred before a total of 2 events from $Y$.
I think I use complimentary counting to solve this problem. The invalid cases I've noted are $X_1, Y_1, Y_2...$; $Y_2, Y_2...$; $Y_1, X_1, Y_2...$ Am I on the right track or is there a more efficient/cleaner method?