Ascribing a probability to an arbitrary future or past event with no information at hand is impossible. It has to be done within a context to be meaningful. Besides that, there are two caveats: certainly if it has happened it has probability 1 (after the time it happened), some events will have probability 0 - such as 'the FTSE will rise 7 points tomorrow AND the FTSE will not rise 7 points tomorrow'.

That said, for any mechanism which ascribes non-zero probability to an event E, E will happen if you take the sample size to be 'large enough'. This is just a restatement of the 'monkeys on a typewriter producing the complete works of Shakespeare' theorem.

There are possible events which have probability 0 too. Stuff that could happen but will not. Like throwing a dart onto the number line and hitting a fraction (or a real world equivalent if reality is continuous).

your intuition is correct, or should i say, formalised to be correct by definition in classical probability, as, informally speaking, as you imply, the probability of getting at least one tail for an unbiased coin in N trials converges to 1 as N -> inf, yet there still remains an infinitesimally small probability of flipping all heads, a result which cannot be directly represented in classical probability due to it defining probability values as lying in the space of real numbers which forbids infinitesimals. This is the likely source of confusion.

Formally speaking, Probabilities are *measures*, which is to say they are functions defined upon sets of outcomes which obey our intuitions of what a probability function should look like. If the set of possible outcomes contains an unlimited number of possibilities the calculus allows us to consistently assign a value of "one" for the entire collection and "zero" for subsets of that set which contain only a finite number of possible outcomes. Such subsets of possibility, although not empty, are said to be of "measure zero" and to never occur "almost surely".

I think the answer depends on how you formalize the question. It also depends on the model of universe you use.

No, why? Here's an infinite sequence of events: 0,1, 0, 1, 0, 1, ... Does 2 ever occur? No. Can you frame a more precise question? The answer to that is no as well. Suppose I flip a fair coin infinitely many times? It's unlikely but perfectly possible that we'll get all heads. Tails may never happen. Remember that ANY particular sequence of heads and tails is just as unlikely as all heads or all tails.

It's simply not true that "everything" must happen. It's also not true that everything must re-occur, or that there would be an exact double of you. Its just false. Something that floats around the Internet with no actual basis in fact.

By the way did you know that if you put a frog in a pot of water and put the pot on the stove and gradually turn up the heat, do you know what happens when the water gets too hot for the frog's comfort? The frog jumps out of the pot. Somebody did the experiment.

Don't believe everything you think.

It's simply not true that "everything" must happen. It's also not true that everything must re-occur, or that there would be an exact double of you. Its just false. Something that floats around the Internet with no actual basis in fact. — fishfry

Let's simplify the universe the way Alec has in his post. The whole universe is just the flipping of coins. It doesn't even matter if they are fair, as long as both heads and tails are possible and all flips are independent. Pick any number of flips and any sequence of heads and tails. The flipper starts flipping the coin. Just to forestall trivial objections - the flipper is able to and will keep flipping for eternity. The floor on which the flipper stands; the table on which the coin lands; the machine that counts the results of the flips; signals when the sequence is correct ; the coin will last forever; and so forth and so on.

In an infinite universe, can you tell me why any particular specific sequence of heads and tails won't eventually show up? Tell me why that sequence won't eventually recur an infinite number of times. Or maybe that's not what you meant.

In an infinite universe, can you tell me why any particular specific sequence of heads and tails won't eventually show up? Tell me why that sequence won't eventually recur an infinite number of times. Or maybe that's not what you meant. — T Clark

Why must it? Suppose I flip infinitely many times. Why can't every flip be heads?

Do you understand that all heads is exactly as likely as any other particular sequence of flips?

Why must it? Suppose I flip infinitely many times. Why can't every flip be heads?

Do you understand that all heads is exactly as likely as any other particular sequence of flips? — fishfry

In my version, I required that the total number of flips and the required sequence of tails and heads had to be specified in advance. If I set the number of flips to 1,000,000,000 to the 1,000,000,000 power and the required sequence to all heads. We turn on the flipper. As soon as it hits a tail, it starts over. In an infinite universe, eventually that number of heads will come up.

In an infinite universe, eventually that number of heads will come up. — T Clark

You keep saying that but there's no reason it should be true. You might get all tails. Or all heads. Any particular sequence is just as likely as any other. Do you understand that point? It's like picking a real number from the unit interval. Convert the number to binary, that gives you an infinite sequence of 1's and 0's which we can interpret as heads and tails. If you pick a number at random (sometimes called "throwing darts at the real line") one bit pattern is just as likely as any other. At each coin flip the odds are 50-50 that you'll get heads. The first flip, the tenth, the trillionth. The odds are always the same. (Also goes for an unfair coin). So why couldn't we flip all heads? Or all tails? Or alternating heads and tails? Every single individual pattern is exactly as likely as any other. No law says that there must eventually be a head or a tail.

Every finite sequence of heads will occur with probability 1. However this is not true for the infinite case.

You keep saying that but there's no reason it should be true. You might get all tails. Or all heads. Any particular sequence is just as likely as any other. — fishfry

Yes, I do understand that each sequence of some particular number of flips has the same probability. Why would that change anything. Instead of all heads, let's pick alternating heads and tails. Or for number of flips n, we'll use the first n values in the decimal expansion of pi, so 3 heads, 1 tail, 4 heads, one tail, five heads, nine tails......

Maybe we're missing what each other are saying. Do you agree, there can be an infinite number of flips?

Every finite sequence of heads will occur with probability 1. However this is not true in the infinite case. — Meta

Yes, I agree. But even in an infinite universe we will never reach the end of an infinite sequence of numbers.

You can't be so sure. Zeno's paradox shows that for example.

Maybe we're missing what each other are saying. Do you agree, there can be an infinite number of flips? — T Clark

Yes. And there's no reason they can't all be heads. You think there's some law of nature that says that eventually there must be a head. But why is that? It's clear that on the zillionth flip, after all preceding flips are heads, the odds are still 50-50 (or whatever the unfair coin's odds are) and that zillionth flip could perfectly well be a head.

I agree that this is unlikely. That's because the event "at least one tail" has many ways of happening, but "all heads" only has one. But each WAY of getting at least one tail is just as unlikely. So you're being fooled by psychological clumping. "At least one tail" includes all the infinitely many different ways you could get at least one tail. You could get a tail on the first flip, or the second, dot dot dot.

But there's no fundamental reason that you couldn't get all heads. It's unlikely. But so is the event of your birth, or someone winning the lottery, or Trump becoming president. Unlikely things happen all the time. In fact every particular thing that happens is incredibly unlikely.

You are confusing something being unlikely with its being impossible. And that's the fallacy in your argument.

You can't be so sure. Zeno's paradox shows that for example. — Meta

To the extent that Zeno's paradox shows anything, that's not it.

Are we really going to turn this into a ZP discussion?

Zeno's paradox "shows" that an infinite number of events can happen in finite time.

Edit: No this wont turn into a ZP discussion but you seem to be struggling with understanding basic concepts.

That's what she said.

Zeno's paradox "shows" that an infinite number of events can happen in finite time. — Meta

Nonsense. [You must know that else why the quotes around the "shows?"] But please, someone start a separate Zeno thread.

For the original topic ... consider the set of all possible infinite bitstrings. One of those bitstrings is 00000. It exists, same as any other one. If we reach into a bag containing all the possible bitstrings, it's possible -- but unlikely -- that we'll pull out the all-zero bitstring. But it's possible. We might pull out some other bitstring. That would be unlikely too! That's all I'm saying.

There are possible events which have probability 0 too. Stuff that could happen but will not. Like throwing a dart onto the number line and hitting a fraction (or a real world equivalent if reality is continuous). — fdrake

That is an interesting point that you've brought up. As you and others have mentioned, probability doesn't seem to make sense when we bring in the infinities. Although we would normally take events with probability 0 to represent impossibilities, that isn't necessarily the case here. In a way, I guess that isn't all that surprising, given that alot of our maths don't make sense when dealing with infinities.

Another way of thinking about the probability of infinite sequences is to reject the assertion that it is meaningful or even possible to linguistically refer to a single infinite sequence. For why should we assume that our finite use of language actually 'picks-out' any single infinite sequence when we appear to talk about it, given that we have not constructed it?

In other words, rather than interpret {H,T,...} to denote a particular yet unspecified infinite sequence of coin tosses beginning with the prefix Head-Tail, lets interpret it to directly represent the entire set of infinite coin toss sequences beginning with H,T.

The advantage of this way of thinking to my mind, is it that it resolves the paradox of our standard theory of probability without us having to change it. We can say that we assign probability one to the entire set of infinite binary sequences simply because our problem is binary and sequential, and yet we can defend our assignment of probability zero to any finite set of infinite sequences simply because we are not interested in these unobservable sets in practice and because they are unthinkable.

This interpretation of classical probability is entirely in line with how it is used in practice, for example when predicting the stock market, where {H,T,...} might represent all of the information a trader might have a time t=2 but where there is no purpose for applying probability to finite sets of infinite sequences that require infinite information.

Even if we extend the rules of probability to assign infinitesimally small probabilities to individual sequences, it turns out, at least on some forms of non-standard analysis, that we will end up describing more than we bargained for. For example we will end up describing sequences of coin tosses that are *even longer* than mere countable infinity, and our original problem resurfaces on an even larger level.

Zeno's paradox was needed to show we can't state for sure that an infinite chain of events is impossible. In fact the only solution I know to Zeno's paradox uses infinite sums and that an infinite number of events can happen in finite time. I dont want to talk more about the paradox since the message of my comment is crystal clear.

Back to the OP. I don't think that "everything" can be formalized properly. But I think if you weaken the claims then you can get a positive answer. If you say "happen 100%" instead of "bound to happen" and mean "local" or "finite" events (whatever this means) when saying everything (and not global events like the whole life of the multiverse) then the answer may be yes, depending on how the universe is modelled.

I think the unintuitiveness of the quantitative behaviour of infinity is something isolated to the folk-mathematics idea of it. Infinity isn't just well understood in mathematics, it's essential.

Zeno's paradox was needed to show we can't state for sure that an infinite chain of events is impossible. In fact the only solution I know to Zeno's paradox uses infinite sums and that an infinite number of events can happen in finite time. I dont want to talk more about the paradox since the message of my comment is crystal clear. — Meta

again, in line with my previous post, one can question whether zeno's paradox is actually conceivable, for it assumes that it is meaningful to talk about infinite and arbitrarily small divisions of a substance, and yet we have never constructed such a thing either on paper, nor with a machine, nor have we ever made arbitrarily small perceptual discriminations.

One cannot say, "yes but zeno's paradox is mathematically conceivable", because again, our mathematical accounts are finite and cannot necessitate the infallible and infinite interpretation required of it.

All we are doing with Zenos paradox is brainwashing ourselves by using the principle of induction.

"I have shown i can keep slicing a cake" means nothing more "I sliced a piece of cake several times until I contented myself with the idea that i can slice the cake again"

Yes we can say that induction is bad, mathematics is bad, we know nothing about the real world, or that the world doesnt even exist. We can disagree in a lot of things. But I still stick to math and science. And I say that Zeno's paradox is mathematically conceivable. Is the mathematical interpretation the absolute and ultimate interpretation? Probably not. Can concieve the structures of Mathematics? No. Does it matter? No. Because Mathematics seem to work.
The mathematical interpretation of the paradox is the only one logically consistent with Newtonian mechanics. So "logically" I dont know why should I deny the possibility of an infinite chain of events. Of course we will never know what the ultimate truth about matter is. But still Maths provides the best answers.

I am really out of the discussion about that off topic.

I am really out of the discussion about that off topic. — Meta

As long as you insist on confusing math with physics, people are compelled to push back. Contemporary physics does not allow for infinite divisibility of matter or time. The question isn't even meaningful since there's a certain point past which we can't measure space or time. Math does allow infinite divisibility, but math isn't physics. I suspect you know this, and I'm not sure why you are pushing this line of argument.

I think the unintuitiveness of the quantitative behaviour of infinity is something isolated to the folk-mathematics idea of it. Infinity isn't just well understood in mathematics, it's essential. — fdrake

Sure, I'm not disagreeing with the fact that we do understand infinity. My point was really about how our finite understanding of probability doesn't transfer over to cases of the infinite, which seems to be the lesson we should take for infinity in general.

All this divisibility thing depends on the definition of time and space (in my opinion). As you can see I prefer the theoretic over the applied or observed aspects of science but I dont want to confuse math with physics. It all started with me trying to give an example using Zeno's paradox but it seems like mentioning the paradox pulls WAY too many triggers on a philosophy forum.

And as I mentioned earlier there are many different models of the Universe on which we can base our calculations.

And as I mentioned earlier there are many different models of the Universe on which we can base our calculations. — Meta

I can live with that. You made an interesting point earlier. In Newton's world we can take space to be Euclidean 3-space. As I understand your idea, you are speculating on what would be true if the universe did happen to be continuous and that we could arbitrarily divide space and time.

In that case we could take a convergent infinite series as true about the world. Calculus would genuinely be a solution to Zeno's questions.

Is that what you are saying?

My "idea" in the first place was to provide an example of why "even in an infinite universe we will never reach the end of an infinite sequence of numbers" is not trivially true for me.
To give an answer to your question: yes, this is what I wanted to say. The mathematical solution to the paradox (in classical mechanics) is an example.

es we can say that induction is bad, mathematics is bad, we know nothing about the real world, or that the world doesnt even exist. We can disagree in a lot of things. But I still stick to math and science. And I say that Zeno's paradox is mathematically conceivable. Is the mathematical interpretation the absolute and ultimate interpretation? Probably not. Can concieve the structures of Mathematics? No. Does it matter? No. Because Mathematics seem to work.
The mathematical interpretation of the paradox is the only one logically consistent with Newtonian mechanics. So "logically" I dont know why should I deny the possibility of an infinite chain of events. Of course we will never know what the ultimate truth about matter is. But still Maths provides the best answers. — Meta

I'm not doubting what we can "know", I'm only doubting what we can mean to ourselves mathematically.

I don't see how Zeno's paradox can be represented in mathematics, since neither calculus nor logic can literally represent a super-task.

In calculus we might write part of a geometric series:

{0, 1/2, 3/4, ..., 1}

But what could it mean to say that the dots "..." represent an "infinite" number of steps?

Is it really the case that "..." is an abbreviation for missing numbers here? Or was it just a sign we invented, along with a rule that if we write "..." then we can allow ourselves to write the limit of the geometric series?

Likewise the rule of induction tells us "given any line divided into finite segments, and given a rule which we grant ourselves to break line segments in smaller segments , then we can permit ourselves to say "the line is infinitely divisible ". But how do we jump from here to the conclusion that "the line really has an infinite number of segments"? For the rule of induction has only given us a definition for what "the line is infinitely divisible" means. And it means nothing else than to say "i have a line of finite segments, and a rule for further subdividing them"

I dont see your point. You say using induction in an example from mechanics is just plain wrong yet your argument in your first post is based on the abstraction of tossing a coin infinitely many times. Where do you want to get with this?