Problem: You have a sequence of N steps that must occur in order. Each step is "unlikely" in any given time period (say, <10%). However, you know that all N steps happened to successfully complete in some trial. In that case, what is the likelihood that the first step completed in the first 1/N section of the total time? How does it change, if the first step is much, much less likely than each of the remaining steps (e.g. 0.01% vs. 10%)?
Edit 7/28/18: There seems to have been some confusion in the comments about the problem formulation. Let me try to add more detail: imagine that there are 10 billion time slices (e.g. "years"). In each time slice, there is an independent and very low probability that some event will occur. For the first possible event, the sum of the probabilities of each year, over the first billion years, is about 10% (or maybe 0.01%) of at least one success. That is, given a billion independent trials, there is a 10% chance that one of the trials will succeed. There are actually 10 billion trials available, in one experiment. However, there is a linear sequence of 10 different "unlikely" events which all must occur, and event N is not even possible until event N-1 has already succeeded. We observe, in one of these experiments with 10 billion trials, that all 10 events in the sequence did in fact occur. The question is: what is the probability that the first event succeeded in the first billion years (1/10th of the time available)? How is that probability conditional on the absolute probability of the first unlikely event (e.g. 10% in a billion years, vs. 0.01% in a billion years)?
Background: This came up in a discussion about the Drake equation and the Fermi paradox. The universe seems to have some kind of unknown Great Filter, because, given our intelligent civilization on Earth, it seems that such a civilization should have already spread to the whole galaxy ... but there doesn't seem to be any evidence out there that the galaxy is already teeming with intelligent life.
So the question, with regard to humans, is whether this Great Filter is behind us, or in front of us. Has human civilization already gotten very very lucky at some step? Or is there great danger of extinction yet ahead in our future?
To help answer this question, you want to estimate the probability of some unlikely steps that human civilization has already passed through in the past (the Drake equation). The formation of life is a key chain. Abiogenesis (life from nonlife) is a critical step. If you look at the timeline of life on earth, it seems that the first life emerged very, very quickly (on geologic time scales) after it might have been first possible (when the molten earth cooled down, the oceans formed, etc.).
Abiogenesis seems to have taken "only" a few hundred million years. Whereas, some other similarly critical steps took much longer: Eukaryotes (cells with a nucleus) took about 2 1/2 billion years. Complex life (Cambrian explosion) took another billion years after that. Evolution didn't especially select for intelligence either; the dinosaurs and other life dominated for 500 million years, without significant intelligent gains until only the last tens of millions of years.
So, just looking at the "life on earth" part of the Drake equation, we have a series of steps like: abiogenesis, eukaryotes, multicellular, complex life, intelligent life. Each one has some unknown low probability of randomly succeeding in any given year (given that the previous steps have already completed). You can perhaps think about it, roughly, as Earth being about 5 billion years old, and human civilization requiring 5 "unlikely" steps in sequence, where each step has some random chance that adds up to something like a 1-10% chance of succeeding in any billion year time span (given that the previous steps have succeeded).
So here's the real question: we know that abiogenesis happened on Earth "quickly" (a few hundred million years). We're wondering whether abiogenesis is a likely candidate for the Great Filter. Is it, potentially, much much much less likely than the other steps in the Drake Equation? Intuition suggests that if it happened on Earth "quickly", that is evidence that the step is not an especially unlikely one.
However, the Anthropic principle disrupts this intuition. We only have this one example (success on Earth). So all we know is, no matter how unlikely each step, they happen to have all succeeded here. And they had to happen in order.
So at last, the real question: what can we conclude about the likelihood of abiogenesis, compared to the other Drake equation steps, given that it has been observed to have happened "quickly" on Earth? Can we conclude that we have some weak evidence that the abiogenesis step is probably "easier" than the other steps? Or, in contrast, does statistics tell us that, given that the whole sequence did in fact succeed, we have learned essentially nothing about the relatively likelihood of each individual step? That, even a (relatively) very very unlikely first step, would still be expected to have been observed succeeding in about 1/Nth of the total time available, conditioned on the fact that all N steps did in fact succeed in this case?
What are the chances that abiogenesis might still be the universe's primary Great Filter, despite the fact that it appears to have happened very quickly on Earth?
Edit 9/16/2018: Robin Hanson seems to have addressed this in his Great Filter essay. In the Technical Appendix at the end, he writes: "[C]onditional on success, all hard steps have roughly the same distribution over durations, regardless of how hard they are."
Edit 9/16/2018: Hanson's more detailed math paper explaining the calculations in the solution.