In time domain $0\rightarrow T$, there are two independent  events $A$ and $B$.
$B$ follows Poisson Process with density $\lambda$. It's easy to get $P_B(t)$ which denotes $P_B(N(\tau+t)-N(\tau)\geq 1)$ in a centain time duration $t$.

$P_A(t)$ denotes $P_A(N(t)-N(0)\geq 1)=P_A(N(t)-0\geq 1)$. $P_A(t)$ means the probability that $A$ happens after a certain time duration $t$ from time $0$. $P_A(0)=0, P_A(+\infty)=1, P_A(t_1)\leq P_A(t_2)\ under\ t_1< t_2$ obviously.

If we don't know what $P_A(t)$ is exactly, we just know there is a $P_A(t)$.

Can we answer the following question with only $P_A(t)$ and Poisson Process $B$?
If not, is more information about $A$ needed? 

What is the probability $P(A\rightarrow B)$ that in time duration $0\rightarrow T$ event $A$ happens and then event $B$ happens(both events happens and $B$ happens after $A$). How to represent $P(A\rightarrow B)$ with $P_A(t)$ and any information about Poisson Process $B$?