Let $(P_t)_{t \geq 0}$ be a Poisson process with $\lambda >0$. Thus, we know that for $t,s \geq 0$ $$P_0= 0$$ $$P_{t+s}-P_s \sim \text{Poisson}(\lambda t)$$
I want to calculate $\mathbb{P}(P_{t+s} = n | P_s = k)$ for $t,s \geq 0$ and $n,k \in \mathbb{N}$. It holds that $$\{P_{t+s} = n\} \cap \{P_s = k\} = \{P_{t+s} - P_s = n-k\} \cap \{P_s = k\}$$ which gives non-overlapping time intervals. Thus, $P_s$ and $P_{t+s} - P_s$ are independent.
I compute $$\mathbb{P}(P_{t+s} = n | P_s = k) = \frac{\mathbb{P}(P_{t+s} = n \cap P_s = k)}{\mathbb{P}(P_s = k)} = \frac{\mathbb{P}(P_{t+s} - P_s = n-k \cap P_s = k)}{\mathbb{P}(P_s = k)}$$ $$ = \frac{\mathbb{P}(P_{t+s} - P_s = n-k ) \cdot \mathbb{P}(P_s = k)}{\mathbb{P}(P_s = k)} = \mathbb{P}(P_{t+s} - P_s = n-k)$$
And here is my question. Which one is correct? $$\mathbb{P}(P_{t+s} - P_s = n-k) = \frac{\lambda^{n-k}}{(n-k)!}e^{-\lambda t}$$ or $$\mathbb{P}(P_{t+s} - P_s = n-k) = \frac{(\lambda t)^{n-k}}{(n-k)!}e^{-\lambda t}$$