There are two events involved, say event A and event B. I want to know the probability of event B conditioned on the event A. The relation between the two events are as follows. We can not talk about about event B if event A does not happen, for example, we can not talk about the probability of death of a person if that person is not born yet. Moreover, once A happened, the events A and B are basically independent. Well, almost independent. For example, once a person is born, the probability of that person's death only has relation with the specific time when that person is born, not with the event of the birth of that person. I hope I explain it clearly enough. Any directions for future references are greatly appreciated. Many thanks for your time and attention.
"B cannot happen until A happens" is the same as 'B is contained in A' or 'B is a subset of A' or 'B is equal to A'. (I'm thinking in terms of Venn diagrams)
When one event is a subset of another, they cannot be independent.
But you can still think of this as two circles intersecting (one for birth and another for time) and the common area to be the event of death (assuming a particular region of time can cause death). Then you can find the probabilities. [here I have three variables- birth, death and time]
I do not have much knowledge about time being incorporated as such in conditional probabilities
PS. I wanted to post this as a comment, but don't have the reputation yet!