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

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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.

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