What is the proper specification/classification for estimating # of occurrences until last event in a series of dependent events?

I don’t believe this nice answer that in case a) presents a negative binomial and in case b) a scaled/shifted chi-squared is what I’m asking, but I very well could be wrong.

Say I have events x1, x2 and x3 which do not necessarily have the same probability of occurring. In order for event x3 to occur, x2 must occur. In order for event x2 to occur, x1 must occur. I think each event is distributed as a geometric random variable. Therefore, E(x1) = 1/p1, but I know that E(x2) != 1/p2 and E(x3) != 1/p3. Individually yes, but the way I’ve implicitly defined those events, it’s actually E(x2|x1) and E(x3|x1, 2), right? If I assume the probabilities are independent, I can just get E(x3) = 1/(p1p2p3).

This "feels" like survival analysis, but I never formally studied it and from what I have seen (briefly), it generally looks like you model time until a single event.

I vaguely remember exposure to markov chains but I’m having a hard time identifying the process that the one I’m thinking may fall under. I ultimately plan to model a more complicated, dynamic process whereby some individuals may "drop out" before getting to x3 based on different distributions of x#_i (sorry for the lazy notation – the idea here is that individuals all "competing" for the same x1, x2, x3 are not drawn from identical distributions).

I appreciate any guidance here and potential references that reflect this class of problems. Very excited to learn more!