How is it possible to win one-in-infinity odds?

[lIllIlIIIl]

TL;DR Summary

Theoretically, it's possible to win one-in-infinity odds. How?

Imagine a roulette wheel with an infinite amount of numbers. Every number on the wheel has a one-in-infinity chance of being selected. Every time the wheel is spun, one number wins those one-in-infinity odds. How is this possible? Isn't one-in-infinity basically zero? It's infinitely far from the potential maximum, making it infinitely small. How is zero different from something which is infinitely small? Please help.

 

[PeroK]

TL;DR Summary: Theoretically, it's possible to win one-in-infinity odds. How?

Imagine a roulette wheel with an infinite amount of numbers. Every number on the wheel has a one-in-infinity chance of being selected. Every time the wheel is spun, one number wins those one-in-infinity odds. How is this possible? Isn't one-in-infinity basically zero? It's infinitely far from the potential maximum, making it infinitely small. How is zero different from something which is infinitely small? Please help.

There is no distribution where a countably infinite number of outcomes have the same probability. The best you could do would be something like:

The probability of outcome 1 is $\frac 1 2$;
the probability of outcome 2 is $\frac 1 4$;
the probability of outcome 3 is $\frac 1 8$;
etc.

In this case each outcome is possible, but the outcomes become increasingly unlikely.

 

[FactChecker]

TL;DR Summary: Theoretically, it's possible to win one-in-infinity odds. How?

Imagine a roulette wheel with an infinite amount of numbers. Every number on the wheel has a one-in-infinity chance of being selected. Every time the wheel is spun, one number wins those one-in-infinity odds. How is this possible? Isn't one-in-infinity basically zero? It's infinitely far from the potential maximum, making it infinitely small. How is zero different from something which is infinitely small? Please help.

As @PeroK says, if you are talking about COUNTABLY infinite, it is possible to give every individual number a positive probability as long as the sum of the probabilities is 1.
Another question is when there are UNcountably infinite numbers, like the Real line from 0 to 1. In that case, there is no way to give every number a positive probability. But given a uniform distribution on [0,1], some number may result from an experiment. In that case, any pre-defined event like $\{ X = 0.5 \}$ has a zero probability. Yet, some event, $\{ X=r \}$ where $r \in [0,1]$ will result from an experiment.

 

[PeroK]

As @PeroK says, if you are talking about COUNTABLY infinite, it is possible to give every individual number a positive probability as long as the sum of the probabilities is 1.
Another question is when there are UNcountably infinite numbers, like the Real line from 0 to 1. In that case, there is no way to give every number a positive probability. But given a uniform distribution on [0,1], some number may result from an experiment. In that case, any pre-defined event like $\{ X = 0.5 \}$ has a zero probability. Yet, some event, $\{ X=r \}$ where $r \in [0,1]$ will result from an experiment.

There is no such experiment.

 

[Dale]

TL;DR Summary: Theoretically, it's possible to win one-in-infinity odds. How?

Isn't one-in-infinity basically zero?

Zero probability isn’t the same as impossible.

 

[Dale]

There is no such experiment.

Are you sure about that? I think you are maybe claiming that we cannot measure an element of a continuum. But I am not sure that is true.

 

[PeroK]

Are you sure about that? I think you are maybe claiming that we cannot measure an element of a continuum. But I am not sure that is true.

If you are talking about a real experiment, then you cannot have infinite precision. There can only be a finite number of possible outcomes. (Maybe countably infinite if you stretch a point)

Moreover, you cannot communicate more than a countable subset of real numbers represented by the computable numbers. For example, you could have a "computable numbers" lottery, where a computer picked a computable number (with the above proviso about unequal likelihoods). But, you couldn't run a real number lottery, because it would be impossible to communicate and compare arbitrary real numbers. They are individually indescribable.

 

[Dale]

There can only be a finite number of possible outcomes.

I am not sure about this. Certainly, if we choose to write our outcome as a machine-representable floating point number, this is true. But science existed before computers.

What if we have an analog measurement that we never digitize? Are there a finite number of possible outcomes of, for example, the centroid of a mark on a paper? It isn’t clear to me.

Anyway, I probably shouldn’t push this point too much. I haven’t thought it through. I just am not confident that indeed there cannot be a real-valued experimental outcome. Since we often successfully model measurements as real numbers I think that I would hesitate to categorically say such an experiment is impossible.

 

[PeroK]

Are there a finite number of possible outcomes of, for example, the centroid of a mark on a paper? It isn’t clear to me.

A better example would be to appeal to the fixed-point mapping theorem. You take a map of London (in my case) and lay it flat on a table. There must be a point on the map that corresponds precisely to the point on the Earth's surface directly below that point.

You could claim that you have picked a point on the map from uncountably many. But, that is still a mathematical operation. In reality, the map and the surface of the Earth are not perfect mathematical objects. At some level of precision, the map is not static. In any case, you cannot communicate the coordinates of the point you claim to have chosen. It would be no use as an uncountable roulette wheel, as you cannot actually produce a real number from your experiment. And, indeed, you can never fully confirm that reality adheres to the mathemetical postulates you have imposed on it.

That, IMO, is the difference between mathematics and physics (in terms of a real experiment). You can have a random variable uniformly distributed on $[0,1]$ mathematically; but no practical experiment can demonstrably produce such a random variable.

 

[PeroK]

PS a related and more obvious example is that you can mathematically have an infinite sequence of random integers. But, you cannot experimentally generate such an infinite sequence. Only a finite sequence of indeterminate length. The Internet is awash with people saying "you toss a coin an infinite number of times", but that cannot be done. You have to stop sometime, even if there is no upper limit to how many times you may toss a coin. You never end up with an infinite sequence.

 

[FactChecker]

There is no such experiment.

That's a bold statement. Are you talking about physically, humanly, impossible or mathematically, theoretically, impossible? Are you talking about the "God's view" result of an experiment or about our human inability to identify the result with infinite precision? I think it's debatable.
Discussion of this might sidetrack the thread and not lead to anything constructive.

 

[FactChecker]

If you are talking about a real experiment, then you cannot have infinite precision. There can only be a finite number of possible outcomes. (Maybe countably infinite if you stretch a point)

Throw a dart at the real line segment, [0,1]. What numbers of Lebesque measure 1 would you rule out as impossible and why? Remember that the remaining countable numbers could have non-zero probabilities in some scenarios. Why are they special?

 

[hutchphd]

You would need a veeery big roulette wheel......

 

[BWV]

There is no distribution where a countably infinite number of outcomes have the same probability.

What about the probability of pulling some number from an interval in ℚ ?

 

[PeroK]

Throw a dart at the real line segment, [0,1]. What numbers of Lebesque measure 1 would you rule out as impossible and why? Remember that the remaining countable numbers could have non-zero probabilities in some scenarios. Why are they special?

A dart is not a mathematical point. It makes a significant three dimensional hole in the board, even on a macroscopic scale.

And the real number line is not a physical object that you can throw a dart at!

 

[PeroK]

What about the probability of pulling some number from an interval in ℚ ?

The rationals are countable. The same argument applies to them as to the positive integers. That is basic probability theory. Mathematically, there is no uniform distribution on a countably infinite set.

The case of the real numbers is more subtle. There exists a mathematical uniform distribution, but that does not map to any physical experiment.

 

[FactChecker]

A dart is not a mathematical point. It makes a significant three dimensional hole in the board, even on a macroscopic scale.

A dart has a center line. Where does that cross? That is a point.

And the real number line is not a physical object that you can throw a dart at!

Ok. I think that you are not allowing any abstract thought on this subject.

 

[PeroK]

A dart has a center line. Where does that cross? That is a point.

At the microscopic scale the point of a dart is a collection of perhaps millions of particles. There is eventually no well defined centre line. Moreover, QM introduces the concept that an individual atom does not have a classically well defined position.

Ok. I think that you are not allowing any abstract thought on this subject.

That's the whole point. Mathematics is abstract, but a physical experiment is not. You cannot directly introduce a mathematical abstraction like a number line and then consider throwing a physical object at it.

A line drawn on a piece of paper is physically a 3D structure of atoms etc. The line itself is not a 1D continuum at the microscopic scale.

 

[lIllIlIIIl]

You would need a veeery big roulette wheel......

I'm a high school freshman, this is the only response I can comprehend and also my favorite.

 

[lIllIlIIIl]

The replies here are amazing, and you all are without a doubt the kind of people that I love to talk to, but my current understanding of infinities is really just that some are bigger than others and that it's a really, really big number. I'm 14 years old. I decided to post this question on a physics forum because I know you guys deal in this kind of equation. I thought maybe one or two people would answer within the month. I did not expect half a dozen physics grads to post here at 2:00 a.m. arguing whether a dart ends in a point. In my opinion, it doesn't, but I also don't think that that's the point they were trying to make. Anyway, all I want to know is, in terms someone in Algebra 1 can understand, how is it possible to win odds of one-in-infinity in this scenario? Am I thinking of zero the wrong way? One is infinitely far from infinity, which I think would make it infinitely small. Is nothingness different from infinitely far from something? I'm very confused. My understanding of numbers is incredibly limited, which is frustrating when I have so many questions like this which require a higher understanding to answer. Please help.

 

[PeroK]

The replies here are amazing, and you all are without a doubt the kind of people that I love to talk to, but my current understanding of infinities is really just that some are bigger than others and that it's a really, really big number. I'm 14 years old. I decided to post this question on a physics forum because I know you guys deal in this kind of equation. I thought maybe one or two people would answer within the month. I did not expect half a dozen physics grads to post here at 2:00 a.m. arguing whether a dart ends in a point. In my opinion, it doesn't, but I also don't think that that's the point they were trying to make. Anyway, all I want to know is, in terms someone in Algebra 1 can understand, how is it possible to win odds of one-in-infinity in this scenario? Am I thinking of zero the wrong way? One is infinitely far from infinity, which I think would make it infinitely small. Is nothingness different from infinitely far from something? I'm very confused. My understanding of numbers is incredibly limited, which is frustrating when I have so many questions like this which require a higher understanding to answer. Please help.

One thing you could try to understand is that if you have a finite number of outcomes, they can all be equally likely. A roulette wheel is an example. But, if you have all positive integers as possible outcomes (that is to say $1, 2, 3, 4 \dots$), then they cannot all be equally likely. They may all be possible outcomes, but you cannot make them all equally likely.

 

[PeroK]

@Dale can you suggest an experiment that would produce any real number (in an interval, say) and prove (or at least justify) why any real number could result.

I assume you accept that most real numbers are indescribable (uncomputable), and the set of computable numbers is countable with measure zero. From that point of view, your experiment could at best claim to have chosen a real number, but could not specify which one. And, in particular, if two such experiments were carried out there would be no algorithmic way to test whether the numbers are equal.

This is a key paradox of the real numbers. We can test mathematically that $x = y$, where $x, y \in \mathbb R$. But, there is no terminating algorithm to check whether two real numbers are equal. Unless you restrict things to the computable subset. IMO, that is a good example of where a simple piece of mathematics (If $x = y \dots$), is not actually physically/algorithmically possible.

 

[BWV]

The rationals are countable. The same argument applies to them as to the positive integers. That is basic probability theory. Mathematically, there is no uniform distribution on a countably infinite set.

The case of the real numbers is more subtle. There exists a mathematical uniform distribution, but that does not map to any physical experiment.

I Googled the proofs of this and of course they are right, but I struggle with why operations allowable on ℝ wont work in ℚ. Take the basic properties of the uniform distribution - is area not defined in ℚ, or does the unit cube have an area of 1? If it does, then ISTM you can compute probabilities for a uniform distribution on [0,1]. you obviously cannot do the MGF or CF as that introduces e, but do these functions define the distribution?

 

[Stephen Tashi]

TL;DR Summary: Theoretically, it's possible to win one-in-infinity odds. How?

Probability theory and even the mathematical theory of statistics gives no answers about whether things are possible or not. Dealing with the possibility of something happening is a matter of applying mathematics to a particular situation. It depends on the science that applies to the situation, not on pure mathematics. The (intellectual) separation between probability theory and its application is hard to appreciate because probability theory is usually taught hand and hand with applications of probability.

A purely theoretical question is whether there can be probability distribution that has an infinite number of equally likely outcomes. Your question suggests you are thinking of a "countable" infinity. In that case, the answer is no. (As others showed, you can have a probability distribution for a countably infinite number of not-equally probable outcomes.]

If you let "infinite number" include a continuum of outcomes (e.g. all numbers between 0 and 1) then the answer is yes. In that case the probability of each individual number is zero. However, probability theory doesn't comment on whether this means that each outcome is impossible versus possible - or possible but not possible to determine with lab equipment etc.

If you are familiar with the physical concept of "density" then you can understand that the mass of a physical object at a point can be zero without contradicting the concept that the object has a nonzero mass density at points and a total mass that is nonzero. Similarly, the probability distribution for a continuum of outcomes is given by a "probability density" function.

 

[FactChecker]

@Dale can you suggest an experiment that would produce any real number (in an interval, say) and prove (or at least justify) why any real number could result.

Don't you think that you have some responsibility to show that the vast majority of numbers are not possible results? Which ones are not possible and which are possible? Do you prefer the ones that only have a finite number of non-zero digits? Why?

 

[lIllIlIIIl]

Probability theory and even the mathematical theory of statistics gives no answers about whether things are possible or not. Dealing with the possibility of something happening is a matter of applying mathematics to a particular situation. It depends on the science that applies to the situation, not on pure mathematics. The (intellectual) separation between probability theory and its application is hard to appreciate because probability theory is usually taught hand and hand with applications of probability.

A purely theoretical question is whether there can be probability distribution that has an infinite number of equally likely outcomes. Your question suggests you are thinking of a "countable" infinity. In that case, the answer is no. (As others showed, you can have a probability distribution for a countably infinite number of not-equally probable outcomes.]

If you let "infinite number" include a continuum of outcomes (e.g. all numbers between 0 and 1) then the answer is yes. In that case the probability of each individual number is zero. However, probability theory doesn't comment on whether this means that each outcome is impossible versus possible - or possible but not possible to determine with lab equipment etc.

If you are familiar with the physical concept of "density" then you can understand that the mass of a physical object at a point can be zero without contradicting the concept that the object has a nonzero mass density at points and a total mass that is nonzero. Similarly, the probability distribution for a continuum of outcomes is given by a "probability density" function.

Yes, this is what I am talking about, thank you so much

 

[BWV]

Yes, this is what I am talking about, thank you so much

You might find this interesting

 

[PeroK]

Don't you think that you have some responsibility to show that the vast majority of numbers are not possible results? Which ones are not possible and which are possible? Do you prefer the ones that only have a finite number of non-zero digits? Why?

Only countably many reals can be described: the computable numbers. if you don't accept that, then there is no point discussing this.

Moreover, the communication of a result can only involve a finite amount of information. You can't communicate a specific infinite sequence unless it can be described in some way. I.e. be computable.

That should be clear.

I still don't think you've sufficiently separated mathematics from experimental physics here. Mathematics can deal with infinite precision through an abstraction. An experiment ultimately comes up against QM at the miscosopic scale. The onus is then on you to say how you can deal with measurements to an arbitrarily small scale.

 

[FactChecker]

Only countably many reals can be described: the computable numbers. if you don't accept that, then there is no point discussing this.

Moreover, the communication of a result can only involve a finite amount of information. You can't communicate a specific infinite sequence unless it can be described in some way. I.e. be computable.

That should be clear.

I still don't think you've sufficiently separated mathematics from experimental physics here. Mathematics can deal with infinite precision through an abstraction. An experiment ultimately comes up against QM at the miscosopic scale. The onus is then on you to say how you can deal with measurements to an arbitrarily small scale.

I don't think that we have any reason to argue or to insist that either position is the only answer. It seems to me that the difference is between what actually happened in God's eye versus what a human experimenter can measure and describe. I consider the former to be the issue in question and you consider the latter. IMO, your interpretation is that an experimental "result" is a range of numbers, determined by the uncertainty of the measurement.

 

[PeroK]

I don't think that we have any reason to argue or to insist that either position is the only answer. It seems to me that the difference is between what actually happened in God's eye versus what a human experimenter can measure and describe. I consider the former to be the issue in question and you consider the latter. IMO, your interpretation is that an experimental "result" is a range of numbers, determined by the uncertainty of the measurement.

I don't see where God comes into it. This is a practical question. We're not talking abstract mathematics, we are talking about an "experiment". That was the term you used in post #3.

 

[FactChecker]

I don't see where God comes into it. This is a practical question. We're not talking abstract mathematics, we are talking about an "experiment". That was the term you used in post #3.

(Edited for clarity) My interpretation is that an experiment has a specific result, known with varying accuracy. "God" (metaphorically) knows the exact result and humans know an estimated result. (I realize that I am not the police with regard to the meaning of the term "experiment".)
Human knowledge of the result can change over time. For instance, monitoring the result of a vaccine shot for its effectiveness and side effects can continue for years. Humans may think they know the result and then be quite surprised by later events.

 

[PeterDonis]

Are there a finite number of possible outcomes of, for example, the centroid of a mark on a paper?

Given that the paper is made of atoms, yes. There are only a finite number of atoms in the paper, and only a finite number of ways to pick out subsets of those atoms as the "marked" ones, from which the centroid then gets computed.

In our models we idealize all this away, but the actual real world experiment has these properties.

 

[PeterDonis]

God knows the exact result

If you are just using "God" as a metaphor here, to signify something like "there is an exact real result, we humans just don't have exact knowledge of what it is", that's fine. But you might want to confirm that.

 

[Dale]

Given that the paper is made of atoms, yes. There are only a finite number of atoms in the paper, and only a finite number of ways to pick out subsets of those atoms as the "marked" ones, from which the centroid then gets computed.

I don’t think that is true. The mark is not a selection of atoms on the paper but atoms of ink or graphite that are deposited onto the paper. Those are not deposited on some uniform lattice. So I am not at all convinced that the possible locations of the centroid is finite.

 

[PeroK]

So I am not at all convinced that the possible locations of the centroid is finite.

It's also a question of being well-defined. It would be simpler to take the position of a single atom at a precise instant. Is that well-defined?