Let $X(T)$ be a Poisson process. What is $ \mathbb{P}(X(t) - X(s) = 1 \mid X(t) = 4) \,? $
I split this up into $\mathbb{P}(X(s) = 3, X(t) - X(s) = 1)$ and found the answer. However, it did not work out to be the solution which is $ \frac{4(t-s) s^3}{t^4} \,. $ I am unsure why I am not able to match the solution. Could someone please help me work this out? Thanks so much!