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extropalopakettle
No offense, but....

 Posted: Mon Oct 04, 2010 10:35 pm    Post subject: 1 This is really a simple probability question, but motivated by thoughts of life having arisen elsewhere in the universe. Assume there are N planets in the universe capable of allowing life to develop. Let P be the probability of life developing on any given one. Assume all N planets are similar, and P is equal for all. 1) What is the maximum value of P (in terms of N) such that there is a 50% or less chance that life developed on more than one planet? 2) For that value of P, what would be the probability of life having developed on exactly one planet? What I'm trying to get at is an idea of how possible it is that we (as life forms) are alone in the universe, without making any a priori assumptions about P (with some backing beyond the "golly, it's such a big universe" variety).
Chuck
Daedalian Member

 Posted: Tue Oct 05, 2010 1:39 am    Post subject: 2 Messing around with a computer simulation for awhile, if I have 100 worlds and give each a 1.678% chance of having life, I get less than 2 with life about half the time and exactly 1 with life about 31.4% of the time. Vastly more worlds would probably take too long to experiment with.Last edited by Chuck on Fri Nov 02, 2012 4:36 pm; edited 1 time in total
Chuck
Daedalian Member

 Posted: Tue Oct 05, 2010 1:49 am    Post subject: 3 With 1000 worlds and a 0.1678% chance of life, I get about half with less than 2 with life and about 31% with exactly 1 with life. I suspect that it will stay around 30% or so for higher numbers.
Chuck
Daedalian Member

 Posted: Tue Oct 05, 2010 1:52 am    Post subject: 4 It's similar for 10000 worlds and a 0.01678% chance of life for each.
milkshake
Daedalian Member

 Posted: Tue Oct 05, 2010 7:05 pm    Post subject: 5 You guys have heard of the Drake Equation and the Fermi Paradox, right?
Chuck
Daedalian Member

 Posted: Tue Oct 05, 2010 9:33 pm    Post subject: 6 I think extro is interested not in how many there are but in the probability that there's just one given that the probability of there being more than one is 50%.
milkshake
Daedalian Member

 Posted: Wed Oct 06, 2010 12:47 am    Post subject: 7 A Rare Earth sort of thing then...
JohnP
Icarian Member

 Posted: Sat Oct 23, 2010 1:23 pm    Post subject: 8 FYI this can be computed explicitly. The chance of life on no planets = (1-P)^N. The chance of life on one planet = NP(1-P)^(N-1) The chance of fewer than two planets with life is the sum of these = [(1-P)^(N-1)]((1-P)+NP) = [(1-P)^(N-1)](1+P(N-1)) = [(1-P)^M](1+PM) (setting M=N-1) Assuming P is very small we can use this: (1-P)^(1/P) = 1/e. Then [(1-P)^M](1+PM) = (1+PM)/(e^(PM)) To make this expression equal 50% we need PM=1.678 as Chuck's simulation showed. So P = 1.678/(N-1). Given this P value the chance of life on one planet = NP(1-P)^(N-1) = NP/(e^(P(N-1))) = 1.678/(e^(1.678)) = 31%, again confirmed by Chuck.
Zag
Tired of his old title

 Posted: Sat Oct 23, 2010 4:57 pm    Post subject: 9 I can't claim to understand the science behind it, but an interesting article I read some time ago claimed that life as we know it could not come from anything less than a third-generation star, which is what ours is. That is, from stars formed in the Big Bang, those stars collapsed and nova'ed to form -- eventually -- new stars; then those collapsed and nova'ed to form stars like ours. The reason is that the first nova forms elements up to iron, but not beyond that, and then two more are needed to form certain other elements that are required for the sort of carbon-chain life that we would recognize. The article went on to say that, with this assumption, life could have formed a couple billion years before it did on Earth, but no more than that. It also said that this assumption excludes a largish percentage of stars from any possibility of life (but only eighty-something percent, really not terribly significant compared to the tiny percentages being assumed for some of the other factors). Again, I can't site a reference, nor do I understand it beyond my understanding that stars are running on nuclear fusion and that it takes extraordinary circumstances (i.e. novas) for fusion to create elements heavier than iron. (... because iron is the most stable nucleus. To make a heavier element you have to put energy into it rather than taking energy out.) Maybe Lepton could explain more. -------- By the way, extro, phrasing the question as "What are the chances of life appearing in TWO places...?" is a little like the guy who brings a bomb on the plane, because the chance of TWO bombs is much less than the chance of one. We already know that life appeared once, so we might as well just exclude it from the calculations altogether.Last edited by Zag on Sat Oct 23, 2010 5:03 pm; edited 2 times in total
Chaz
Vote: Zag

 Posted: Sat Oct 23, 2010 5:00 pm    Post subject: 10 Oh man. I would love to read that article. It sounds right up my alley. Any idea where you read it?_________________The enemy's base is down.
extropalopakettle
No offense, but....

Posted: Sat Oct 23, 2010 9:19 pm    Post subject: 11

 Zag wrote: By the way, extro, phrasing the question as "What are the chances of life appearing in TWO places...?" is a little like the guy who brings a bomb on the plane, because the chance of TWO bombs is much less than the chance of one. We already know that life appeared once, so we might as well just exclude it from the calculations altogether.

My whole point is to try to get some mathematical perspective on what the probability might be that it happened only once. Is there some probability of it occurring such that it's not highly improbable that it would occur once, and also not highly improbable that it occurred only once?
Lepton*
Guest

 Posted: Sun Oct 24, 2010 1:48 am    Post subject: 12 Interesting approach, extopalopakettle. I think the conventional wisdom in SETL circles is simply to assume that there is a really frigging tiny chance of life arising on only one planet: these calculations show that to be true, but the probability is higher than I'd expected. Zag: Exactly. I don't recall the Drake Equation dealing very well with the requirement of a 3rd-generation star. The relevant bit is something like "the portion of stars in the universe that could support life-bearing planets", which is usually taken to mean the portion of stars in the universe that might host planets, with no mention made of the heavier elements being present.
extropalopakettle
No offense, but....

 Posted: Mon Oct 25, 2010 3:46 pm    Post subject: 13 Does it make sense to talk of meta-probabilities here? I mean, we can identify a certain limited range of values for P such that if the probability of life arising on a given planet (with some conditions) is P, then it's not so unlikely that there be one planet (of the many having those conditions) with life, but not more than one. The question is, knowing nothing else, what is the probability that the actual value of P lies in that range?
JohnP
Icarian Member

 Posted: Mon Oct 25, 2010 11:19 pm    Post subject: 14 I'm reminded of this problem: There's three urns with 10 balls each. They have the following contents: Urn A has 1 black 9 white Urn B has 2 black 8 white Urn C has 3 black 7 white The urns are not labeled. You pick a random ball from an urn and the ball is black. What's the probability that you drew from urn A? This problem is answerable because we know the probability in each urn and we assume each urn has a 1/3 chance of being chosen. By analogy we can consider an urn to be a universe and a black ball is a planet with life. We know we're living in an urn. We know there's at least one black ball. We want to know the probability that we're in an "A" urn. BUT in this case we know neither the distribution of possible universes nor the probability of life for a given universe. So I don't know where we can go.
Chuck
Daedalian Member

 Posted: Tue Oct 26, 2010 12:25 am    Post subject: 15 I'd be tempted to say that we're probably in the urn with the most black balls but we can see some of the other balls in our urn and see no other black balls.
Zag
Tired of his old title

Posted: Tue Oct 26, 2010 12:55 am    Post subject: 16

 extropalopakettle wrote: I mean, we can identify a certain limited range of values for P such that if the probability of life arising on a given planet (with some conditions) is P, then it's not so unlikely that there be one planet (of the many having those conditions) with life, but not more than one. The question is, knowing nothing else, what is the probability that the actual value of P lies in that range?

I would hazard that it is probably more likely than any other single number. That is, the likelihood of exactly one planet in the universe having life is higher than, say, fourteen. If the value of P is essentially zero, then it took an infinity of universes to produce any at all, and the chances of a second one in any universe is zero. The vast majority of universes have no life, but there's no one talking about it on any of those.

I don't believe this, of course. But I think that it is more 'likely' than any other specific number.
L'lanmal
Daedalian Member

Posted: Tue Oct 26, 2010 2:32 am    Post subject: 17

 extropalopakettle wrote: Does it make sense to talk of meta-probabilities here? I mean, we can identify a certain limited range of values for P such that if the probability of life arising on a given planet (with some conditions) is P, then it's not so unlikely that there be one planet (of the many having those conditions) with life, but not more than one. The question is, knowing nothing else, what is the probability that the actual value of P lies in that range?

Sadly, this is dangerous for several reasons.

1) You make no statement about independance of chances of life. It is quite possible that life developing on one planet make it much more or less likely to occur on other planets. This is commonly addressed in science fiction (about the seeding of planets), and it could be that our development of radiation and technology influences other planets. But it is probably the least of my objections.

2) You are trying to define a probability that life develops given that the planet is capable of developing life. How do you define "capable of developing life" if life has not indeed developed?

----------------
Edit: On review, my upcoming points 3 & 4 you addressed in your original post.
 Quote: What I'm trying to get at is an idea of how possible it is that we (as life forms) are alone in the universe, without making any a priori assumptions about P (with some backing beyond the "golly, it's such a big universe" variety).

Restricting to "likely" values of P is implicitly making a priori assumptions about P, which is what you said you wished to avoid doing. For a value to be likely after observing exactly one world of life, it would need to be possible and plausable a priori.
-----------------

3) You are intending to calculate P(p=x for x between 0 and 1 | one planet has life). This sounds like a Bayesian argument, where you adjust your perceived probability given a piece of observed emperical evidence.

This is fine to this point, but this philosophy requires you to have in mind a distributions of probabilities before you make your observation. (That is, an a priori probability distribution.) This will necessarily be subjective.... it sounds like the a priori distribution you have in mind is a uniform (0,1) distribution: that each probability of life is equally likely. For instance, 99% chance of life developing would be equally probable as 1% chance of life developing, if we did not know that life developed on exactly one planet. But this is arbitrary, and other people could make equally valid arguments by starting with slightly different assumptions.

The alternate school of probability (frequentist, as opposed to Bayesian) requires you to be able to repeat the experiment to define your probabilities. That is you (or someone or something) would need to be capable of rebooting the universe and measuring if life developed each time.

4) The range of "unlikely" probabilities of life is likely to have a large measure, so cannot be safely ignored even though "individual" probabilities are low. This is the "golfer's paradox": when you hit a golf ball, the odds of it landing whereever it is does are astronomically low, but you know it has to land somewhere.
extro...*
Guest

Posted: Wed Oct 27, 2010 12:34 am    Post subject: 18

 L'lanmal wrote: 1) You make no statement about independance of chances of life. It is quite possible that life developing on one planet make it much more or less likely to occur on other planets.

I'm considering the event of abiogenesis - development of life from non-life. These I believe are independent events.

 L'lanmal wrote: 2) You are trying to define a probability that life develops given that the planet is capable of developing life. How do you define "capable of developing life" if life has not indeed developed?

Well, I would consider any planet that is nearly identical to Earth as it was just when life first developed here (or if it didn't, the argument is easily modified) ... any such planet is among those "capable of developing life". There is some probability that life actually will develop on such a planet. There are chance events involved. Is the probability high or low, I don't know. "capable of developing life" means conditions exist in which life could develop, but not necessarily that it would.

Of course in reality there would not be a division of the planets into those incapable of developing life and those that are all, with the same probability P, capable of developing life. But I think it's a reasonable simplification to get a rough idea where the line of thinking leads.

(may get back to other points later; why no feature to save draft versions of posts?)
L'lanmal
Daedalian Member

Posted: Wed Oct 27, 2010 2:22 am    Post subject: 19

extro...* wrote:
 L'lanmal wrote: 1) You make no statement about independance of chances of life. It is quite possible that life developing on one planet make it much more or less likely to occur on other planets.

I'm considering the event of abiogenesis - development of life from non-life. These I believe are independent events.

I don't think independence is at all necessary. In the case of a Creator, there could be a clear common cause. We know of local and global astronomical phenomena which could have a common influence on many planets, such as local supernovas, fluxuations in neutrino levels, debris left from comets within a single solar system, proximity to pulsars, and the like. From a causality perspective, we have developed technology which emits radiation. And we know of at least one mechanism, "gravity", through which objects can be influenced at great distances.

I agree with you that it is reasonable to assume independence, but I think it needs to be stated.

 Quote: Well, I would consider any planet that is nearly identical to Earth as it was just when life first developed here (or if it didn't, the argument is easily modified) ... any such planet is among those "capable of developing life". There is some probability that life actually will develop on such a planet. There are chance events involved. Is the probability high or low, I don't know. "capable of developing life" means conditions exist in which life could develop, but not necessarily that it would.

So imagine taking a number of copies of early earth, some of which develop life and some don't? Makes sense from a frequentist perspective.

(However, I personally enjoy thinking that we are more likely to find life on a non-earthlike planet than an earthlike. I would like to believe that we have not yet conceived of all the mechanisms by which life can exist and nature will once again surprise us.)
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