Just a quick question regarding two Poisson Processes:
Let $X_t$ and $Y_t$ be two independent Poisson Processes with rate parameters $\lambda_1$ and $\lambda_2$, respectively, measuring the number of customers arriving in stores 1 and 2, respectively. What is the probability that a customer arrives in store 1 before any customers arrive in store 2?
My approach to this problem thus far has been to consider all possible times where store 1 could have a customer arrive, but that gets into dealing with infinity and I'm not so sure that's correct. Mathematically, I'm thinking I should calculate
$$P(X_1 = 1)P(Y_1 = 0) + P(X_2 = 1|X_1 = 0)P(Y_2 = 0) + P(X_3 = 1|X_2 = 0)P(Y_3 = 0) + ...+ P(X_n = 1|X_{n-1} = 0)P(Y_n = 0).$$
Is there an easier approach than the one I am taking? Is the approach I'm taking even correct?