We have $k$ distinct light switches, so let's label them $0$ up to $k-1$. Suppose they are all off initially.
At random time intervals we are simply switching one of the light switches on and off again. We choose the particular switch uniformly at random.
Let $N_t$ denote the total number of times we have done this up to time $t$, and note that $t$ is a continuous time variable.
We further suppose that $\{N_t \}_{t \geq 0}$ is a Poisson process with rate $\lambda$.
Here's my question:
Given we have chosen switch $0$ twice up to time $t$, find the probability that we have $\textbf{only}$ chosen switch $0$ up to time $t$.
I have tried to use the thinning of a Poisson process theorem, but seem to arrive at the concluson that it is not relevant how many times we have chosen switch $0$. But this doesn't make sense, since if we have not chosen switch $0$, then the probability is necessarily $0$, but if we have then the probability must be positive.