Question:
Assume $N_1(t)$ and $N_1(t)$ are two independent Poisson processes with rates
$λ_1$ and $λ_2$, respectively. Show that $N_1(t)$ + $N_2(t)$ is a Poisson Process with rate $λ_1 + λ_2$, and use this to obtain the probability that the first event of the combined process is from the $N_2$ process.
Where I'm stuck: I've shown that $N_1(t)$ + $N_2(t)$ is a Poisson Process with rate $λ_1 + λ_2$. The problem is that when I use this to obtain the probability that the first event of the combined process is from the $N_2$ process, I get a result that differs from what I get when I compute the same probability in a different manner. In particular:
I reason that the first event of the combined process will be from the $N_2$ process if and only if $T^{1+2}_1$, the time of the first event of the combined process, is less than $T^{1}_1$, the time of the first event of the $N_1$ process.
Since $N_1(t) + N_2(t)$ is a Poisson process with rate $\lambda_1 + \lambda_2$, $T^{1+2}_1$ is exponentially distributed with parameter $\lambda_1 + \lambda_2$. And since $N_1(t)$ is a Poisson process with rate $\lambda_1$, $T^{1}_1$ is exponentially distributed with parameter $\lambda_1$.
Therefore:
$$P(\textrm{1st event of } N_1(t) + N_2(t) \textrm{ is from } N_2(t)) = P(T^{1+2}_1 < T^{1}_1) = \frac{\lambda_1+\lambda_2}{(\lambda_1 + \lambda_2) + \lambda_1}$$
However, as a check I also reasoned that the first event of the combined process will be from the $N_2$ process if and only if $T^2_1$, the time of the first event of the $N_2$ process, which is exponential with rate $\lambda_2$, is less than $T^1_1$, the time of the first event of the $N_1$ process, which is exponential with rate $\lambda_1$. Therefore: $$P(\textrm{1st event of } N_1(t) + N_2(t) \textrm{ is from } N_2(t)) = P(T^{2}_1 < T^{1}_1) = \frac{\lambda_2}{\lambda_1 + \lambda_2}$$ And these two results differ. Where am I going wrong?