I get that the probability of event $A$ is $\langle A|A\rangle$.
Given that $A$ has happened what is the probability of B happening at $t$ seconds in the future? My first guess is:
$$ P(A|B) \stackrel{?}{=} \frac{\langle A\vert U(t)\vert B\rangle}{\langle B|B\rangle}$$
where $U(t)$ is the time evolution operator. But isn't this a complex number?
And how is this related to the transition Green's functions $G(A,B)$ i.e. the amplitude going from state $A$ to state $B$. ?
You should think of quantities like $\langle B \vert A \rangle $ as describing probability amplitudes, where the probability is recovered by doing the "norm-squared" action $P(B\vert A) = \vert \langle B \vert A \rangle \vert^2 $. So $\vert \langle A \vert A \rangle \vert^2 = 1$ and resembles $P(A\vert A)$, not your best-guess equivalent $P(A)$.
Language like "happening" is kinda imprecise, but more importantly it dodges the big issue of uncertainty in QM. We might say that given an initial state $A$, the probability of measuring some condition/state $B$ after time $t$ is: $$ P(B\vert A) = \vert \langle B \vert U(t) \vert A \rangle \vert^2 $$
but we didn't know what "happened" without the notion of some measurement (or distinguishable effect) of the time-evolved state (this is addressed in Kyle's link). Use of the time-evolution operator like this implies that $B$ is some observable feature that $A$ is at least partly composed of; i.e. $A$ and $B$ are not simply conditionally linked events.
