This question is related to this question. I'm struggling to get the correct answer from an approach I follow and it doesn't look wrong to me.
The question is simply if we have two independent Poisson processes: $N_1(t)$ and $N_2(t)$ with different rates $\lambda_1 , \lambda_2$ consequently. Find the probability that a first arrival comes from $N2$ before the first arrival of $N1$.
So, if the first event occurs from $N_2(t)$ Poisson process before the first event occurs from $N_1(t)$, which is also a Poisson process. Then simply we start with
$$P(N_2(t) = 1 , N_1(t) = 0) = P(N_2(t) = 1) * P(N_1(t) = 0)$$
Where $t$ is just an arbitrary time interval. However, this doesn't lead to the expected answer which is $\lambda_2/(\lambda_1 + \lambda_2)$. I think there is a flaw and misunderstanding in my approach but I am not able to see what is wrong in my way of thinking?