Poisson Process Thinning Process - Shock Arrival

Question

I don't understand why $P(N=n | T=t) = P(N_1(T)=n-1| T=t)$ and it is not taking $N_2(T)$ into account.

I think it should be $P(N=n | T=t) = P(N_1(T)=n-1| T=t) \cdot P(N_2(T)=1|T=t)$ where $P(N_2(T)=1|T=t) = \lambda p e^{-\lambda p t}$.

Where am I getting wrong?

Answer 1

This question has confused me for a while. I finally understood it myself. This is because $T$ denote the time of failure, in other words, the condition of $P(N=n|T=t)$ already implies $P(N_2(T)=1|T=t)$.