The number of pulses that arrive to a Geiger counter follow a Poisson process of rate "three pulses per minute". Every pulses that arrive is registered with probability $\frac{2}{3}$. Let $(X(t),t\geq0)$ the process that describes, for every $t$, the number of pulses registered in $[0,t]$. Find, for a generic $t$:
1) $\mathbb{P}(X(t)=0)$.
2) $\mathbb{E}(X(t))$.
3) $\mathbb{P}(X(3)=6|X(2)=3)$.
For 1) we have $$\begin{align}\mathbb{P}(X(t)=0) & =\sum_{k=0}^{\infty}\mathbb{P}(N_{t}=k,X(t)=0)\\ &=\sum_{k=0}^{\infty}\mathbb{P}(N_t=k)\mathbb{P}(X(t)=0|N_{t}=k)\\ &=e^{-2t}\\ \end{align}$$
For 2) and 3) i'm stuck.
Could you please explain me how to set up the problem? Thanks in advance for any help!
EDIT: For point 3), why we have
