I saw this problem in Durrett.
Consider two independent Poisson processes $N_1(t)$ and $N_2(t)$ with rates 1 and 2. What is the probability that the two-dimensional process $(N_1(t),N_2(t))$ ever visits the point $(i, j)$?
I wrote this as $P(N_1(t)=i,N_2(t)=j)=P(N_1(t)=i)P(N_2(t)=j)=\frac{e^{-\lambda_1t}(\lambda_1t)^i}{i!}\frac{e^{-\lambda_2t}(\lambda_2t)^j}{j!}$
Is this all that is there to the problem or is there something I am missing?