Poisson process with uniformly distributed random variables

Question

Assume a Poisson process $\{N(t),t\geq0\}$ with rate $\frac{1}{12}$, where $t$ is a random variable that is uniformly distributed over $(2,4)$. Then how to find $P(N(t)=0)$?

So I started with assuming $t=3$, then I know $P(N(3)=0)=e^{-\frac{1}{12}\cdot3}\frac{\left(\frac{1}{12}\cdot3\right)^0}{0!}=e^{-\frac{1}{4}}$. But how should I deal with uniformly distributed random variables?

Answer 1

Hint: Use conditional expectation, $$P(N(t) = 0) = E[ P(N(t) = 0 \, | \, t) ]\,.$$