Can you roll a die and get the same number an infinite number of times?

[octopus]

@Trinopus

How do you add up a bunch of zeros and get one?

The width of a point has a limit of zero, but is not zero. Read up on real topology and neighborhoods. Re-read the fundamental theorem of calculus. If points had “zero” width, calculus wouldn’t work; you couldn’t add up the dxs to obtain a non-zero result.

Any probability that actually equals zero is impossible. Probabilities with limits of zero are what you are trying to describe: they’re as close to zero as God himself can’t tell, but they do add up to the entire dart-board. A bunch of zeroes – even Aleph-One of them – still adds up to zero.

You just refuse to read the math, man. The width of a point or a line are 0. The thickness of a plane is 0. These aren’t limits. And yes calculus would work with points that have 0 width. Especially since the points have 0 width. And then we have an infinite amount of functions that are nowhere differentiable. Weierstrass function - Wikipedia

And if you did that integral (with a sample function of your choice) that I typed which admittedly has terrible formatting you will see that the probability is indeed 0. The upper and lower limits of integration are identical as they should be for a point. Which means the answer is 0 not a limit.

I’m aware of the differences between a “dx” and a point and they are separate concepts and separate math entities. Now the question I must ask is do you think it’s more probable that texts used at MIT and every other college are wrong on this topic when it’s specifically addressed and discussed or that your intuition is at fault?

[octopus]

@watchwolf49

As the number of possible outcomes of the trial approaches infinity, the probability of a specific outcome approaches zero.

The question seems to be whether this means the probability is zero, or is it simply defined as zero based on the definition of a limit. I think the whole notion of an indefinitely small probability is a mathemagical curiosity, and conveys no meaningful information regardless if it can be stated as a zero value. 1/∞ must be defined as some real number otherwise it is not and we wouldn’t be able to add or multiply it.

Sure, physically rolling dice is a limit. A point on a line segment or a point on a circumference or a point in an area is actually 0 and not a limit.

[octopus]

@Indistinguishable

Returning to this, here’s an example of why one might wish to say rolling a die and getting infinitely many 3s has probability exactly zero, and not merely infinitesimally close to zero:

Let X be the probability of getting a 3 on infinitely many rolls in a row. This amounts to first rolling a 3, and then, subsequently, getting a 3 on the infinitely many rolls in a row that follow. Accordingly, we should have X = 1/6 * X.

But from this, it follows by ordinary algebra that X = 0. Not just that X is infinitesimally close to zero, but that X is zero, exactly

Now that’s clever. I haven’t seen that argument before.

[Thudlow_Boink]

@Trinopus

I think this is the problem. The probability of choosing a real number from a finite section of the number line is not zero, but is an infinitesimal with a limit of zero.

@Trinopus

How do you add up a bunch of zeros and get one?

The width of a point has a limit of zero, but is not zero. Read up on real topology and neighborhoods. Re-read the fundamental theorem of calculus. If points had “zero” width, calculus wouldn’t work; you couldn’t add up the dxs to obtain a non-zero result.

I don’t know what you’re talking about. Modern standard calculus doesn’t involve infinitesimals or “adding up the dxs” (at least not in how things are formally, rigorously defined and worked out). Early calculus did, but the contradictions and lack of rigor in this approach led later generations to reformulate things on a solid logical footing that didn’t involve infinitesimals.

Relatively recently, “nonstandard analysis” was developed as a coherent way of formulating things in terms of infinitesimals, but I don’t know enough about it to know whether what you’ve said makes sense within that context, or whether it works to approach probability that way.

[Half_Man_Half_Wit]

@Trinopus

Any probability that actually equals zero is impossible. Probabilities with limits of zero are what you are trying to describe: they’re as close to zero as God himself can’t tell, but they do add up to the entire dart-board. A bunch of zeroes – even Aleph-One of them – still adds up to zero.

Or, perhaps think about it this way. If there is some x > 0 such that you propose it’s the probability of an infinite sequence of die rolls coming up 3, then observe that there is some integer n such that x > (1/6)[sup]n[/sup]; i.e. that the probability of obtaining 3 n times is smaller than x, which was your proposed probability of 3 coming up infinitely often. Hence, it follows that the probability of getting infinitely many 3’s must be smaller than any x > 0 (while being >= 0, as that’s how probabilities are defined); but only 0 fulfills that requirement (within the real numbers).

Or, yet another way: think about a random variable X yielding values x from the (real) interval [0,1]. Then the probability of finding a value between a and b, b >= a, is P(X in [a,b]) = b - a. So again, the only choice for the probability of having X yield exactly x is P(X = x) = x - x = 0.

If that doesn’t help, maybe try the wikipedia article on ‘almost surely’, the concept that an event that has probability 1 doesn’t necessarily happen, which is the converse of what we’re discussing, which clarifies that:

[QUOTE=wikipedia]
If an event is almost sure, then outcomes not in this event are theoretically possible; however, the probability of such an outcome occurring is smaller than any fixed positive probability, and therefore must be 0. Thus, one cannot definitively say that these outcomes will never occur, but can for most purposes assume this to be true.
[/QUOTE]

[Biffster]

How about the reverse question? Can you roll a die and get a different number on each roll an infinite number of times?
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[Isilder]

@Dr.Strangelove

Probably a real mathematician can answer this better, but I’ll give it a shot.

Let’s suppose we pick our number with a 10-sided die, and we want to match some pre-determined number, like 0.222… or pi.

You’ll agree, I think, that the odds of rolling the first digit correctly are 0.1. And then the odds of getting the first two digits right are 0.01, and so on.

You’ll also agree, I hope, that the odds of getting N+1 digits correct are lower than getting N correct.

So suppose I just ask you a series of questions:

• Are the odds (of rolling infinite digits in a row) less than 0.1?

I’d stop right there… you are NEVER rolling an infinite number of times, AND we don’t need to calculate the hypothetical odds for rolling an infinite number of times, because we already know that for any finite number of rolls you or your descendants (or of whoever takes over rolling for you or them) become extinct at, there is a finite, non-infinitesimal probably for the result… it could happen, but only because you reached a finite limit.
The asymptote does head toward zero, which implies that the probability at infinity is zero, but this is arguably an artifact of achieving the impossible first… which just shows that if its impossible the probability should be zero… its impossible… no chance… If you are considering an infinite number of independent events… stop right there, thats impossible…

Don’t do cosmology with this result though, its probably not meant to control the behaviour of “the universe” either.

[Isilder]

Its purely metaphoric or allegorical.

Suppose dad is talking to his teenage children “you do not walk on the streets at night, call me and I’ll come and get you!”.

the sidewalk is metaphorically rolled up.

Its just idiom.
Take your girlfriend to a bar, and its just you and her in there… Right ? SO you feel a force telling you to leave… you could say, the bar was metaphorically locked… its your feeling of being banned, not the actual practical difficulty with walking on the public right of ways.

So what if there was literally rolled up pavement, its not an expression that is still in use due to habit and continuity of use… the places with pavement that rolls up were at the time rather more lively than the places that currently “roll up the pavement” metaphorically now.

[Biffster]

You could also do a trial test and extrapolate. If it is not possible in any practical sense to, say, roll a three one hundred times in a row, let alone a thousand or a million, you could demonstrate by extension that it is definitely not going to be possible for infinity times. The odds of rolling a 3 se about five to one at the best of times. Even with those odds, the likelihood of NOT rolling a three in one hundred tries is still pretty much zero. As the number of tries increases, the law of averages takes over. At some point, say, a thousand tries, you are much more likely an even distribution of all possibilities from 1 through 6, give or take a view. The higher the number of attempts, the more even the distribution (unless the die is loaded of course). The usefulness of debating an impossibility is an odd focus of time and energy in my opinion.
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[watchwolf49]

Root 8 over pi.

[watchwolf49]

Missed the edit window … my mistake … I was supposed to wait until Trinopus posted …

So, what are the “chances” of a moderator deleting my post #130?

[Half_Man_Half_Wit]

There’s two ways to interpret this question. One is as a question about probability theory—in this case, it’s perfectly well possible to talk about infinitely many die rolls and the probability of coming up a certain way an infinite number of times: the theory is there, it’s well developed, and there’s a correct answer, which is 0.

The other is to interpret it as a question of performing a given physical action, throwing a die, an infinite number of times. Now, as things go, this is simply impossible: there is no die that can be thrown that often, nor can any infinite series of actions be performed in finite time (barring some oddities that might arise, such as in Malament-Hogarth space times, where two observers can arrange to meet after an infinite proper time has elapsed for one of them, while the other’s clock shows a finite reading, things like Thompson’s lamp etc.). Interpreted this way, the question really makes no sense: you can’t throw a die infinitely often, so what happens if you do that is a moot point.

Many participants in this thread simply chose to interpret the question in the first way—which makes it, after all, a perfectly sensible question with a GQ answer.

In fact, depending somewhat on which theories of physics ultimately turn out to be right, such considerations may have an impact even in the physical world—eternal inflation, for instance, posits that there exists an infinite number of universes; there, questions of probability, such as the probability of a given universe having a certain value for one of its physical constants, say, the cosmological constant, indeed do turn out to depend on considerations of probability on infinite sets.

So I think just to say ‘you can’t throw a die infinitely often’ sells the issue short: there are interesting issues well worth a discussion to consider if one allows for the possibility of infinite outcome sets.

[Buck_Godot]

I only skimmed the responses so far but I don’t think that the definiative answer is posted here, so here we go:
**
The way real mathematicians think about this problem**

In probability theory, a random variable (such as the roll of a die) is actually viewed as a map from a set of possible outcomes onto a measurable space with total measure equal to one, wherein the probability of a set of events it equal to the measure of the subset of the measure space that that set of events maps to.

As a particular example of this we can consider an infinite number of rolls of a 10 sided die to map to a number whose digits is those successive rolls. So the roll of an infinite number of 6’s maps to 0.6666666… =2/3. No need to actually roll and infinite number of dice or approach anything, its just a 1 to 1 mapping between an infinite die roll and the number [0,1] (letting 0.99999=1).

So if I want to know the probability of getting a particular event say of rolling number between 1/sqrt(3) and 1/sqrt(2) it will be the area between these points. If I want to know the probability of rolling a number in the Smith-Volterra-Cantor set it will be 1/2.
So when I ask what is the probability of getting an infinite number of 6’s will be the width of the number 0.666… which is zero.

“But,” you complain, “that is true of every number so doesn’t that mean that no number can possibly come up?”

This is the equivalent of saying that since every number on the interval from [0,1] has width 0, then the width of the entire interval from 0 to 1 must have width 0. But thanks to calculus we know how to add together an infinite number point with width 0 and still get a positive area.

[octopus]

@Biffster

How about the reverse question? Can you roll a die and get a different number on each roll an infinite number of times?
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Each sequence has the same probability.

[Czarcasm]

@octopus

Each sequence has the same probability.

Given an infinite number of rolls, it is impossible to roll an infinite number of times.

[Indistinguishable]

@Buck_Godot

If I want to know the probability of rolling a number in the Smith-Volterra-Cantor set it will be 1/2.

FWIW, you linked to the Cantor set (which has measure 0), instead of the Smith-Volterra-Cantor set (which has measure 1/2).

[Indistinguishable]

Thus, this link is the relevant one: Smith-Volterra-Cantor set.

(Yes, there is a broader sense of the term “Cantor set”, in which the Smith-Volterra-Cantor set is a Cantor set, but in this broad sense, Cantor sets can have all kinds of measures including, in the canonical ternary example, zero)

[Buck_Godot]

Thanks,

I first was going to use the Cantor set as an example but then changed my mind (since a set of measure 1/3 was more interesting than one of measure 0), Unfortunately I forgot to change the link.

[Trinopus]

@watchwolf49

Root 8 over pi.

“Don’t ever tell me the odds!”