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Suppose we have an urn that contains infinitely many balls with different colours. Black balls are very common, and white ones uncommon. We do not know exactly how uncommon white balls are. However, we are told that if we draw a ball from the urn, the probability of drawing a white ball is strictly greater than 0. Suppose we take N draws from the urn. Probability theory requires that the probability of drawing at least one white ball approaches 1 as N approaches infinity.

Similarly, given that we exist, we know that the probability of life emerging on a given planet is strictly greater than 0. Assuming the universe is infinite, is it certain that there exist other extraterrestrial life in the universe?

  • To continue your analogy: in our universe, the ball is already drawn. So we're actually talking about a conditional probability: what is the probability that other life, and or sentient life in particular, exists elsewhere in the universe, given that it already exists here on Earth? i don't believe there's any undebateable way to assign that probability. You might like to read about the Drake equation as a starting point, and go from there. – Alexis Dec 2 '17 at 5:12
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    "Similarly, given that we exist, we know that the probability of life emerging on a given planet is strictly greater than 0." Do we? – H Walters Dec 2 '17 at 6:15
  • The certainly that life does NOT exist anywhere but on Earth is largely a religious view, which largely correlates with the view that the universe is essentially either a random collection of atoms or an exquisitely designed habitat for Earth. If, on the other hand, you think of the universe as a living thing, then it puts things in a new perspective. en.wikipedia.org/wiki/Entropy_and_life (Also, see Gaia) – David Blomstrom Dec 3 '17 at 5:04
  • Given what we know about biology, evolution and so on its pretty certain that amongst the billion stars of a billion galaxies there is life; we simply cannot be certain of finding it. The distances are simply too vast. – Mozibur Ullah Dec 3 '17 at 11:36
  • It's impossible to draw a red ball from the urn. And there's no particular reason anything must happen with absolute certainty in an infinite universe. You should not confuse probability with certainty. Infinitary probability theory doesn't work that way. – user4894 Dec 3 '17 at 21:41
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Let's assume there are an infinite number of planets.

Similarly, given that we exist, we know that the probability of life emerging on a given planet is strictly greater than 0.

Assuming that by know here you mean that it directly follows, then surprisingly enough, no, we don't know that.

Let's go to the marbles and I'll explain why:

Suppose we have an urn that contains infinitely many balls with different colours. Black balls are very common, and white ones uncommon.

I propose a minor change; let's say the balls are equally common. We draw a black ball with probability 1/2, and a white ball with probability 1/2, those being collectively exhaustive and mutually exclusive, regardless of the number of draws.

Now let's draw an infinite number of marbles. Since I imagine doing so can get pretty boring, I propose we play a game as we do so. I'll supply you with one infinitely dense chocolate bar, labeled along its length equally with all of the real numbers from 0 to 1. I'll draw the marbles. Before each draw, you cut the chocolate bar exactly in half along the length. After the draw, you get to eat one of the halves; if I draw a black marble, you eat the larger numbered half (leaving the lower one for the next turn). If I draw a white marble, you eat the smaller numbered half.

To give you an idea; after three draws, you will have eaten 7/8 of the chocolate bar. There will be 1/8 remaining, but there are 8 different ways this can happen. I propose a simple observation; after some number of draws, you will have eaten some amount of the chocolate. However long that uneaten bar is, the a priori probability you will have wound up with that specific portion of chocolate is the same as its length.

We're now ready to play the game an "infinite" number of times, somehow. By doing so we'll wind up inching towards some point-thin slice of uneaten chocolate labeled with some real number. That slice of chocolate necessarily has a zero width (in particular, were I to challenge you to name any positive width, no matter how small; I can tell you the finite number of draws that sliced the bar smaller than that).

Now note my previous observation; the length of the slice of uneaten chocolate is the same as the probability that that particular piece would be selected. The length of your point thin slice is 0. So the probability that it would get selected is 0. But we will select some piece by playing this game; it's impossible not to. It follows, then, that there's something counterintuitive going on with probability once the infinite gets involved; there's such a thing as an event that has a probability of exactly 0, but is nevertheless possible. Formally, an event that has a probability of 0 but can still occur is said to "almost never" occur, which is distinct from being impossible (though impossible events also have probability 0). The complement (an event that has probability 1 but can still not occur) is said to "almost surely" occur, which is distinct from being certain (though certain events also have probability 1).

So let's backtrack now. Suppose there's a probability 1/2 on each planet that life will be there (a "pure" probability, like "flipping a coin" or more aptly "drawing marbles from an infinite urn"; not a frequentist one). Then it's no surprise we're here; but given the infinite number of other planets, almost surely some other planet will have life on it. But it's still possible none of them do (this is equivalent to selecting 1/2 on the bar; i.e., the first time you pick a white marble, then never do again).

So in summary, given life exists on our planet, but under the assumption that there are an infinite number of planets (or even not strictly), all we can establish with certainty is that life is possible (that is, we're here, therefore it's possible we're here). That doesn't even imply that the probability of a planet having life is greater than 0. But even if the probability were greater than 0, we cannot be certain there's life elsewhere... even if that probability is high; at best we can claim that there's almost surely alien life.

That's still a lot; I wouldn't bet against almost surely.

(FYI, the above maintains a non-Bayesian view of probability; using a Bayesian approach would clash with the proposed notion that we don't know what the probability is).

  • "Similarly, given that we exist, we know that the probability of life emerging on a given planet is strictly greater than 0" does not mean that the probability of it happening again is strictly greater than 0. Maybe "god did it." – Daniel Goldman Dec 2 '17 at 15:49
  • Also, so long as p(E) is greater than 0, and each trial is independent, then P(at least one success) tends toward s 1 as trials tend towards infinity. – Daniel Goldman Dec 2 '17 at 15:52
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You are essentially asking for the probability of at least one alien planet having life, given an infinite number of planets. This question is basic probability question.

Suppose the probability of a planet (currently) having life is p, where p is strictly greater than 0. According to current theories on how life forms, this assumption is reasonable. There is a basic formula for calculating the probability of at least one success (here a success is life forming on an alien planet). That formula is 1 - p(none). In cases where each trial is independent, that formula boils down to 1 - p^n where p is the probability of success and n is the number of trials.

Since p is strictly greater than 0 and less than 1, the limit as n tends towards infinity for 1 - p^n is 1. We could say that the probability of life not forming anywhere else is infinitesimal, but we do not work with such concepts in probability theory. Probabilities are real numbers.

In either case, the result implies that so long as we are correct in our assumption that p is strictly greater than 0, it is almost guaranteed that alien life exists on some other planet.

Real Probabilities

I am going to add a full section on probabilities. It is annoying, but 0 and 1 do not mean impossible and necessary, when we're talking about dealing with scenarios about infinity. They mean almost never and almost always. We cannot know if 0 means actually impossible or 1 means absolutely necessary, using current probability theory. The only option is to extend probability theory using hyperreals, which has been done. Regardless, it does not change the understanding of the result in this case, by too much. Is it absolutely certain that there is alien life? No. It is however almost certainly the case, again, assuming p > 0.

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A probability of 0 is not the same as logical or metaphysical impossibility. In your urn example, for any sample size N, it's logically possible to draw a sample where all N balls are black, whatever the probability of drawing a white ball. In the same way, (given that it's possible to draw an infinite sample) it's logically possible to draw an infinite sample where every ball is black. This event has probability 0, but it's still possible. (I think this is similar to H Walters' point.)

However, certainty isn't the same thing as possibility. Many academic philosophers think it's reasonable to be certain that probability 0 events will not happen even when those events are possible. Or, equivalently, it's reasonable to be certain than probability 1 events will happen even when they're logically or metaphysically contingent. The IPCC — the international body that produces climate change assessments — used the term "virtually certain" to characterize findings that have probability 99%-100% (link).

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