I was waiting at a train station when I thought of this problem. Suppose I know the average rate of arrival for trains coming in from the left and going out on right from the station, say $\lambda_1$, and an average rate of arrival for trains coming in from opposite direction, say $\lambda_2$.
What is the probability that if I wait for a time period T, I will see at least one pair of trains crossing right in front of me?
I also thought of a solution, but I am not sure if I am on the right track. I designed a new Poisson Arrival process with a rate $\lambda$ = 1/LCM(1/$\lambda_1$,1/$\lambda_2$). Am I doing this correctly?