Conditional Probability of Poisson Process

Question

I have seen many questions on here regarding the conditional probability of the Poisson Process however I have not seen an option for the opposite

i.e \begin{align} P(N_T = n | N_t = m) \end{align}

where $T > t > 0$ and $n > m > 0.$

Could someone help me understand this please?
Am I missing something obvious?

Sorry for the poor format this is my first post.

Answer 1

Due to the memorylessness of the Poisson process,

$$ P(N_T=n\mid N_t=m)=P(N_{T-t}=n-m\mid N_0=0)=\mathrm e^{-\lambda(T-t)}\frac{(\lambda(T-t))^{n-m}}{(n-m)!}\;, $$

where $\lambda$ is the rate parameter of the Poisson process.