E.T. Life and Probability

[extropalopakettle]

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]

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.

[Chuck]

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]

It's similar for 10000 worlds and a 0.01678% chance of life for each.

[milkshake]

You guys have heard of the Drake Equation and the Fermi Paradox, right?

[Chuck]

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]

A Rare Earth sort of thing then...

[JohnP]

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]

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.

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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.

[Chaz]

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]

[Lepton*]

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]

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]

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]

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]

[L'lanmal]

[extro...*]

[L'lanmal]