# A Poisson process question

I saw an old post here, claiming that for a Poisson Process $X(t)$:

$P[X(t) - X(s) = 1 \mid X(t) = 4]=\frac{4(t-s) s^3}{t^4}$

Am I missing something essential about stochastic processes, probability or the Poisson process? If I would have tried to solve it, I would have said:

$P[X(t) - X(s) = 1 \mid X(t) = 4]=P[4 - X(s) = 1]=P[X(s) = 3]=\frac{e^{-s\lambda}(s\lambda)^{3}}{3!}$

What am I doing wrong in the previous line?

Using the definitions, one sees that $$P[X(t) - X(s) = 1 \mid X(t) = 4]=\frac{P[X(t) - X(s) = 1, X(t) = 4]}{P[ X(t) = 4]}=\frac{P[4 - X(s) = 1, X(t) = 4]}{P[ X(t) = 4]},$$ and this would equal your suggestion $$P[4 - X(s) = 1]$$ only if the numerator could be rewritten as $$P[4 - X(s) = 1]\,P[ X(t) = 4],$$ that is, if $X(s)$ and $X(t)$ were independent. But $X(s)$ and $X(t)$ are not independent.