Then pick a framework. You don’t get to mix and match the math you’re playing in. The rest of us appear to be talking about standard Cantorian infinities. If you’re using something else, you have to say so or else this entire thread is totally futile.
ultrafilter had already explained this mathematically when you posted so you have the answer. So I need to guess what your question is. The clue for me is saying “at any given point.” That’s a finite statement. No matter where your given point is, it is effectively zero to infinity. It seems as if you’re asking when a very large finite number starts behaving infinitely, sort of like asking when quantum processes start acting like macro processes. I don’t think there’s an answer to that question. Infinities act differently from finite quantities because they are qualitatively as well as quantitatively different. Your intuitive feel for probability no longer operates. But if this is the case, you’re asking a philosophy question more than a math question.
I’m having some difficulty understanding the OP’s question, so let’s ask explicitly. Is your question simply about the difference between “probability of 1” and “guaranteed to happen”? [And similarly the difference between “probability of 0” and “guaranteed not to happen”?]
I don’t have a problem with events with zero probability happening as such; I know I can choose a number out of infinitely many numbers even though the probability for me to choose exactly that number is zero (though there are days where I feel a bit uncomfortable with the concept in practice, since I don’t have any actually infinite pool of numbers to choose from; at best, I could maximally choose a number which it takes me the rest of my life to recite, and among those I could realistically choose, there’s probably a strong bias towards those among the first few billion). But it seems to me that the fundamental conflict isn’t resolved by what you (and Thudlow) said (unless, of course, I misunderstood you): if it is indeed possible to choose some C-less string, which I take you to be saying, then what does it mean for C to have probability of 1/3?
In other words, why are the more probable outcomes actually more probable? How is it that reality (nearly) always chooses a string which approximates the correct distribution of events occurring better and better the longer it gets? Couldn’t it equally well choose one where that isn’t the case, confounding the hell out of people in casinos everywhere?
I’m perfectly well able to grasp how probabilities behave with repeated sampling, and all is indeed well if you look at this from a ‘bottom-up’-perspective: with each roll of the die, there’s a certain probability for you to not roll a six, and in each succession of rolls, the probability of there not being a six gets smaller with the length of the succession. Viewed from there, all is well. But now, what if mother nature, before the rolling commences, grabs out of her grab-bag of successions one of infinite length, which doesn’t happen to include a six anywhere (call this the ‘top-down’ perspective). This has probability zero, as you said, but so has grabbing a succession with the correct distribution of sixes. Now, as the player throws the die, we can watch him grow at first irritated, than perhaps a little spooked, and then freaked out like hell if he continues to not throw sixes. And there’s the dilemma: we can usually rely on our bottom-up notion of probability being correct; however, viewed from the top-down perspective, there does not appear to be a reason for this if there’s no preference for mother nature to grab the right succession out of her bag – thus, that successive die rolls don’t follow the rules of probability should perhaps be as common as that they do.
(I don’t believe that I’m right with what I’m saying, by the way; I just can’t figure out where I’m wrong.)
That’s probably what this comes down to, yes.
That’s more or less what I’m asking – take my button example: you rig the whole thing such that A, B, and C have the same probability, heck, you may even start a trial to determine if you’ve done it right and measure that all three lamps light up roughly one third of the times the corresponding button is pressed, and then you start your actual experiment – and C never lights up again!
On the standard interpretation of probability: It could. However, with probability 1, it won’t (that is, with probability 1, the limiting frequency of your favorite letter in the chosen string will equal its probability, this being the “strong law of large numbers”).
Here’s your error. The set of all sequences with no sixes has probability zero because there are constraints on an infinite number of events. But the set of all sequences with the approximately correct number of sixes does not–in fact, it has probability one, although the math you’d need to show this is a little bit more sophisticated than what I’ve discussed so far.
No, I don’t believe it is. I think that my question is actually most concisely stated as: Given an infinite amount of time, is it true that anything that can happen, will happen?
If the answer to this question is ‘yes’, then I don’t believe I have a problem at all – however, among other things as per the argument I’ve given in the OP, it actually seems to me that the answer is ‘no’.
Apologies to anybody I’ve already snapped at in this thread, particularly Sage Rat and Exapno Mapcase. You’re trying to help, and my snark was undeserved. I should probably get away from the computer for a few hours, I’ve had a long day.
Anyway, sorry again, and thanks to everyone so far!
The answer is “no, but the probability that it doesn’t happen is zero” assuming that we’re discussing a sequence of independent events. That’s very different from a simple “no”.
Ah, ok. Well, as Chronos said, it’s up to you to formalize what you mean by “can happen” and similar alethic language. But it would not be unusual to pick a formalization on which the answer is “No”, and for the exact reasons you gave in your OP; even without bringing probability into it (and why should we? Probability, on the standard account, says nothing about “can” and “will”, instead talking at most about “non-zero probability” and “probability of 1”). We can allow, for example, “Sure, the actual sequence came out AAAAAAAA…, but it was still possible, at each juncture, for B to come out instead; it just didn’t actually happen.”
At least, we can allow this up until you end up picking a more restrictive formalization of what “possible” means. But you just get out what you put in; it’s up to you.
Well, and also assuming “that can happen” is replaced with “that has non-zero probability of happening”. [That is, just as your rephrasing turns possibilistic “will” into probabilistic “almost surely will”, we need also to similarly rephrase possibilistic “can” into probabilistic terms]
If you restrict yourself to talking about sequences of independent draws from a stationary distribution, which is what the OP seems to be doing, then you get definitions of all those terms for free. In general, the answer depends on what exactly you mean by those words, but I think that here the meanings are pretty clear.
If it were true that “anything that can happen, will happen, given enough time,” then you’re guaranteed eventually to get a completely satisfying answer to your question if this thread goes on long enough.
I think there’s the beginnings of an a-ha brewing in the back of my mind. Indistinguishable and ultrafilter, thanks to both of you, I think that was exactly what I needed to hear to resolve my dilemma.
That’s assuming that I can, in fact, be satisfied – perhaps excessively charitable given my conduct in this thread so far. But thanks!
I don’t know what exactly you mean when you say “you get definitions of all those terms for free”, but alright. In any case, it seemed that lack of clarity about what was meant by those words was the core of the question.
Frankly, I think it’s best to keep saying “There needn’t be any particular correspondence at all between possibility and probability*, so if the question is solely about possibility and necessity, it’s of no use to bring up probability distributions, just possibility distributions (and, conversely, if the question was meant to actually be just about probability, then why look at possibilities?)”, over and over.
*: Unless we impose one. But so it is with everything…
With all due respect, Half Man Half Wit, I really don’t understand your position.
If an event can happen, then it is possible to wait any arbitrary number of millennia with that event not occurring. But for infinity to elapse, and that event has still not occurred — doesn’t that prove that the event was actually not possible?
I think you are confusing “infinity” with “a ridiculously long time”. “Infinity” means that at some point, all conditions existed, in all possible combinations. For all those situations to have occurred, yet your Event did not occur, seems (to me) a clear demonstration that you were mistaken when you thought that this event can happen.
Well, no, that’s not what infinity means. At least, it’s certainly not the only commonly accepted meaning. A set can be infinite without containing everything.
In fact, you’d be quite hard-pressed to have a set which does contain everything. Under all the standard formalisms for set theory, such a set is impossible. You could, though, have a set containing all integers, or something tame like that.
Okay, I surrender. Here are a few things which we might agree will categorically never happen, even if this universe lasts forever:
[ul]
[li]Even if this universe lasts forever, a citizen of Antarctica will not be the first human on the moon.[/li][li]Even if this universe lasts forever, no one will ever draw a pentagon with only four sides.[/li][li]Even if this universe lasts forever, a rock will not spontaneously grow arms and write a copy of the Declaration of Independence.[/li][/ul]
Perhaps the OP should be rephrased more rigorously. Exactly how do we define the phrase “Anything that can happen”. What is meant by “things which can happen”?
Hmmm… Upon rereading the actual OP, I see that perhaps he did explain himself clearly enough, by using the example of events A B and C, which supposedly have an equal chance of occurring, yet the OP suggests a world in which an infinite time might elapse before C ever occurs.
Okay, here’s my response: What is the procedure for determining the odds of something occurring? I don’t know if there is any rigorous procedure, except for experience. For example, the odds that a coin will land heads up is NOT exactly 1/2. The two sides may be awfully close in weight, even to the tolerance of our best measurements, but it is still possible that for a given sort of coin, ten trillion random flips will tend to have six more heads than tails. In the real world, we’d never notice that, but in the context of the OP, “given enough time”, yes, we surely would. It is only after infinity has passed that we can be sure whether the odds were 50.000000000000000001% or if the odds were truly and exactly 50%.
So too, it is only after infinity has passed that we can know whether the event in question is something which can happen (as proven by its having happened), or if it is something which cannot happen (as proven by its not having happened).
It depends on whether we’re talking about math or reality.
Coins that have a 1/2 probability of coming up heads, or events A, B, and C that are equally likely to occur, are mathematical abstractions, like points, lines, and circles are.
Don’t forget that when dealing with infinity, any possible event, no matter how improbable, will happen an infinite number of times. The typical example being a class in which all students are named Jesus Christ. Infinity is also the end at which all things even out.
