# What is the likelihood that the first of N unlikely steps occurs in the first 1/Nth of the total time, given that all N steps succeed?

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.

• The background is TL;DR. However, your problem statement is not well-defined. You have not given any information on the time required to complete steps; therefore you cannot make any inference about the time taken for the first step conditional on the completion of the series. If you would like to be able to infer the latter, you need to specify some information about the times taken for steps.
– Ben
Jul 21, 2018 at 6:52
• I can make up hypothetical times if you think it would help. Abiogenesis takes "a year" to complete -- when it works. In a billion year time span, that one successful year will occur in ONE of those billion trials, about 0.01% of the time. Does that provide you enough information about "the time required to complete steps"? (Also, there are 9 other steps in the sequence, each of which also take "a year" to complete -- when successful -- but which have about a 10% chance of getting a successful year in any billion year timespan, conditional on the prior steps already having succeeded.) Jul 22, 2018 at 15:21
• Are you saying that the probability of an event's occurring, given that it hasn't yet occurred, is the same for every year? (Doesn't seem very plausible for the events relating to factors in the Drake equation, but we could go with it.) But that would describe a process that always completes, given enough time, so what are you conditioning on? That the process completes in exactly so many years? That the process completes within so many years? (Thinking about the Anthropic Principle, perhaps it's that the process completes before all the stars've gone out, or something like that.) Jul 26, 2018 at 11:26
• Yes, I was using a crude model that the probability of success is the same in every year, for these kinds of random "lucky" events. Doesn't necessarily mean that the process "always" completes: for example, if an event has a 1% annual chance in a year, then the chance that it doesn't happen at all in a century is 99%^100, which is about 1/3 of the time. But yes, the time limit is the current age of our sun (4-6By) and its eventual full lifetime (8-10By). The question is what we can learn from our one example here. Jul 27, 2018 at 14:58
• Could you please edit the question then to include this clarification of which probabilities you want to calculate? Jul 27, 2018 at 22:24

This is a problem concerning how to measure evolution. The answer is perhaps not what you want to hear. The chances for having a Star Trek universe, where we have intelligent beings of almost the same intellectual prowess at the same time is small. Consider the examples we have here on earth. Apes do not talk for lack of coordination of vocal cords in the hind brain, but they can learn sign language and a chimp usually develops intellectually faster than a human baby until 2 years of age. And that is about as close as it gets, we are in the ape family, so to speak. If we consider other mammals, dolphins have echolocation and ultrasound that we literally cannot hear, so that if they were even speaking English, we would not know. So far we have what happens when you change the rules for communication between close species; cluelessness.

Now consider that one interpretation of intelligence is more having the right information at the right time, for example, to solve a problem, and consider that we are nearing a "singularity" or a cascade of problem solving ability that is species wide, and not individually centered. Newton said that he saw further because he stood on the shoulders of giants, and so do we all. No matter how much we may lament "group think" we are stuck with it, for example, this site is (on its better days) a co-op. I did not invent English, and there are those who claim that I can barely use it.

So, we have the speed of thought which can be very different, we have gross intelligence that can be at very different levels, we have group thought and social learning, language conventions, different methods of communicating including different physics for relating information and rapidly changing circumstances for intelligence to adapt to.

How then can you tell if there is an extrasolar intelligence very close to us? Chances are, you would not even notice because the rules of the game could easily be so different that the "Star Trek" type overlap is near zero. In fact, if there were an intelligent species of fish living in one of Earth's ocean's volcanic vent regions that communicates with chemical excretions or light flashes, we just would not have enough in common to pass much in the way of information between ourselves and those fish.

However, if those fish build a star ship, that we might just notice. But how do you suggest striking up a conversation? When Drake wrote his equation, he put in some pretty vague ideas. Stipulating that an alien intelligence exists, and having any common ground for a conversation are vastly different propositions. The former is likely, the latter is not.

I think that the development of life is not a much of a filter at all. It is not tool usage, beavers make huts, dam streams, make ponds, and change the environment hugely. The great filter is likely communication. To communicate with us, a prototypical alien would have to have a level of intelligence close enough to ours to make that worth while, ten time less intelligent than we are would make it not worthwhile and if they were ten time more intelligent than us it would not be worth their while. If they weigh more than 1000 times more than us or 1000 times less that could be an impediment to communication. If they think more that 10 times slower or faster than us that could be an impediment. If they do not use the spoken or written form that would be an impediment. If they use higher or lower frequency sound waves to communicate that would be an impediment. If they communicate with chemical signals like insects do, that would be an impediment. If they use photoluminescence to communicate like fireflies or some fish, that would be an impediment. If they have incompatible biology, are non-biological having evolved into a machine race, good luck. I could go on and on. The great filter is indeed us, we are not expecting to find what is really out there.

• You're discussing the difference between mere intelligent species, vs. a starfaring civilization. That is indeed yet another step on the Drake equation, but it wasn't the one I was focusing on. (As you say, if volcanic fish invent starships, it wouldn't be too difficult to notice.) But I was trying to focus just on abiogenesis, and the observation that it happened "quickly" on Earth. Does this, or does this not, provide data that abiogenesis is "easy" in the universe? That's a simple (?) probability question. Jul 22, 2018 at 15:13
• Abiogensis is most likely very common, although the theory itself is not necessary as biologic material travels due to collisions. I would be surprised if biologic material from the Earth, including some RNA/DNA was not already spread all over the solar system and beyond, and the converse also seems likely; that biologic material has found its way to Earth many times. However, we are not talking about higher life forms that do not travel that easily. However, if life were an encyclopedia, intelligent life would be printed on a single sheet of transparent plastic, in invisible ink.
– Carl
Jul 23, 2018 at 4:57
• You're discussing other approaches to explaining the Fermi paradox. I might have a lot of opinions about other Fermi options as well, but in this question I was trying to focus on one simple statistics question. What have we learned about the universe's probability for direct abiogenesis on earth, given that it happened "quickly" in geologic time? Does math tell us that means it is "easy" ... or does math instead tell us that we still have almost no information about its absolute probability? Jul 24, 2018 at 14:04
• Early life falling to earth is not "abiogenesis", which instead means the creation of life from non-living materials. What you're actually saying is that abiogenesis on earth was not necessarily required; it could have happened elsewhere, and then simply fallen onto earth, rather than occurring on earth itself. I get that. But you still haven't answered my question: IF abiogenesis DID happen on earth, and IF it happened "quickly", then what does probability tell us about how "hard" the step is? You telling me that earth life could have started without earth abiogenesis, is not relevant. Jul 26, 2018 at 2:16
• I understand what you're saying. I am NOT trying to answer an actual question about abiogenesis. What I am trying to do, is answer a math question, and then USE the answer of the math question, in order to inform my understanding of some possibilities of part of abiogenesis. But nobody seems to be helping me understand the math question, so I remain confused about how to interpret that part of the geologic record. (And I don't understand why you think my original question was unclear. I don't know the answer, but the question itself seems perfectly straightforward to me.) Jul 28, 2018 at 22:53