How to compute conditional probability for a homogeneous Poisson process?

Question

Let $N$ be a homogeneous Poisson process with intensity $\lambda$. How do I compute the following probability: $$P[N(5)=2 \, | \, N(2)=1,N(10)=3]?$$

Answer 1

You know that exactly $2$ events occurred between $t=2$ and $t=10$. These are independently uniformly distributed over the interval $[2,10]$. The probability that one of them occurred before $t=5$ and the other thereafter is therefore $2\cdot\frac38\cdot\frac58=\frac{15}{32}$. The intensity $\lambda$ doesn't enter into it, since you know the numbers of events.