Characteristics of Poisson Process

Question

I am having trouble setting up a model for this seemingly simple problem:

Given two independent Poisson processes, what is the probability that on of the two reaches level two before the other reaches level one? The answer ought to be 1/2, but how can I show this?

Thanks in advance for the help

Answer 1

Basic approach. Use the fact that the joint process is memoryless, so that you're basically just counting sequences of coin tosses. Thus, HH and TT both satisfy the condition, and HT and TH do not.