Conditional Poisson processes with Multiple conditions Let $\lbrace N(t)\rbrace_{t\geq 0}$ be a Poisson process with intensity $\lambda = 3$.
Compute $$P\left[N(6) = 2 \,|\, N(8) = 4, N(3) = 1\right].$$
I understand when there is only one condition i.e. $P\left[N(6) = 2 | N(8) = 4\right]$. Since that is $$P\left[N(6) = 2 , N(8) = 4\right]/ P\left[N(8) = 4\right] = P\left[N(6) = 2 ,  N(8)-N(6) = 2\right]/ P\left[N(8) = 4\right].$$ But I am unsure on how to go about doing this one 
Any tips on where to begin? Thanks

You can handle this using the definition of conditional probability, stationary increments and independent increments.
$$
\begin{align*}
P\left[N(6) = 2 \,|\, N(8) = 4, N(3) = 1\right] &= \frac{P[N(6) = 2,N(8) = 4, N(3) = 1]}{P[N(8) = 4, N(3) = 1]}\\
&=\frac{P[N(3) = 1, N(6) - N(3) = 1, N(8) - N(6) = 2]}{P[N(8) = 4, N(3) = 1]}\\
&=\frac{P[N(3) = 1]P[N(6) - N(3) = 1]P[N(8) - N(6) = 2]}{P[N(8) = 4, N(3) = 1]}\\
&=\frac{P[N(3) = 1]P[N(3)= 1]P[N(2)= 2]}{P[N(8) = 4, N(3) = 1]}\\
\end{align*}
$$
I'll let you finish it up from there.
