I How to convince someone that an event repeating an infinite number of times does not guarantee every outcome?

[sairoof]

TL;DR Summary

How to convince someone that an event repeating an infinite number of times, does not guarantee every outcome?

I'm having this discussion about an anime character. (Spoilers for jojo's bizarre adventure part 5 ending) the main antagonist is put inside a loop where he dies and gets revived in a different place, and time. In the comic the character died from a dog bite, car crash, and a surgery gone wrong, implying that there are an infinite number of ways that this character dies.

Here is where the discussion starts. I know for a fact that not every outcome is guaranteed, for example: this character dies from a heat stroke. It is possible, but not guaranteed, and when I try and convince people of it they usually just reply with, it happens an infinite number of times, so every possible way is guaranteed, and we start a loop of arguing back and forth for nothing.

So, how would I convince someone that not every death to this character is canon to the story?

 

[jbriggs444]

I do not hold out much hope for trying to convince the mathematically uninitiated with an English language argument.

However, one approach might be to explore the distinction between "probability zero" and "cannot happen". See the Wiki article on "almost surely".

The probability of rolling a die infinitely many times and never getting a six is zero. But that does not mean that it cannot happen.

 

[FactChecker]

You need to distinguish between the physical ability of something happening versus the odds or probability of that thing happening.

If the number of possible outcomes are uncountably infinite, I agree with you. It is only possible to get a countable number of results and there are more possible results than that. So many results never happened.

On the other hand, if there are only a countable (infinite or finite) number of possible outcomes, I disagree with you. If you have any outcome with a possibility greater than zero, the odds of the results avoiding that outcome forever is zero. It might be physically possible, but there are zero odds (probability) of it never happening.

 

[jedishrfu]

Why not use a simple coin toss?

Everything seems the same for each toss and yet its not a perfectly predictable event.

Or a more nuanced example, is dropping a ball on the ground. The drop is perfectly predictable but where the ball winds up after a few bounces is not.

 

[WWGD]

Maybe Borell Cantelli would be involved? https://en.m.wikipedia.org/wiki/Borel–Cantelli_lemma

 

[DaveC426913]

On the other hand, if there are only a countable (infinite or finite) number of possible outcomes

I'd say, intuitively, that there are not an infinite number of discrete ways to die. It seems to me, that, with a finite structure such as a human body, with a finite number of atoms, it has to be so. While the number of ways those atoms could be separated from each other (or merely mixed together in non-functional ways) such that metabolism ceases may be very large, it is not infinitely large.

 

[sairoof]

I'd say, intuitively, that there are not an infinite number of discrete ways to die. It seems to me, that, with a finite structure such as a human body, with a finite number of atoms, it has to be so. While the number of ways those atoms could be separated from each other (or merely mixed together in non-functional ways) such that metabolism ceases may be very large, it is not infinitely large.

In this case, the circumstances of dying is what changes with each time.
So he might die from his skull crushed once by a car, and once by lifting heavy weights.

 

[FactChecker]

In this case, the circumstances of dying is what changes with each time.
So he might die from his skull crushed once by a car, and once by lifting heavy weights.

You can imagine a really huge number of ways that a person could die, but I think they would still be finite. Infinity is much greater than any finite number.

 

[PeroK]

TL;DR Summary: How to convince someone that an event repeating an infinite number of times, does not guarantee every outcome?

I disagree with much of what you have been told. There are two main types of infinity. The first is countable infinity. An example of this is the whole numbers $1, 2, 3, 4, \dots$. The second is uncountable infinity. An example of this is the real numbers. There is a simple, beautiful proof that the real numbers are uncountable.

However, you can only do something a finite numbers of times. That number might be unbounded, but it cannot be infinite. Mathematically, you can repeat a process an infinite number of times. But, you can only repeat your story a finite number of times. There is no way to produce an infinite number of stories.

If we are talking about something in the real world (or imagined about the real world), then we have a finite number of possibilities.

If we are talking about mathematics, then we can have a countable or uncountable infinite number of possibilities. However, if by a story we mean a sequence of words or a sequence of images (each of which is itself finite), then the number of possibilities is countable. It's possible mathematically to define an infinite story. Or, an infinite sequence of stories.

An example is to have the character toss a coin repeatedly. And the story is simply the sequence of heads and tails. Practically, that story or sequence must terminate at some point - even if there is no point at which it must terminate. Mathematically, that story or sequence can be assumed to continue indefinitely.

This is what I assume we are talking about here. In this case, if the coin has two sides (heads and tails), then it must land heads at some point in the story and tails at some point. If not, then the relevant probability is zero by definition.

In other words, if heads can happen (probability of heads not equal to zero), then it must happen.

My answer would be that everything that can happen must happen. In the sense that everything with a non-zero probability of happening in your story must happen at some point in your repeating story.

Note that the repeating part is important. For example, we could have a story where the character tosses a coin once to start. If it lands heads, then it lands heads thereafter and likewise if the first toss lands tails, then it lands tails thereafter. In this case, there is probability of 1/2 that the coin lands heads in your story, but unless that happens on the first toss, then it will never happen.

Finally, probability theory can be extended to the concept of a continuous probability distribution on an uncountable subset of the real numbers. However, real numbers and continuous probability distribtions are strictly mathematical things. And, trying to conceive of an uncountable number of stories becomes a tricky mathematical exercise. Not least because all but a countable subset of the real numbers are indescribable. I.e. most specific real numbers cannot be described or communicated in any way.

If you move into the strictly mathematical world of continuous probability distributions, then there is a sense in which everything has a probability of zero. This is effectively noting the the length of each real number is zero. The problem comes if you conclude things about actual probability distributions based on that observation. That's been discussed on here before and the Internet is full of people making dubious claims about probabilities in this respect.

 

[sairoof]

I disagree with much of what you have been told. There are two main types of infinity. The first is countable infinity. An example of this is the whole numbers $1, 2, 3, 4, \dots$. The second is uncountable infinity. An example of this is the real numbers. There is a simple, beautiful proof that the real numbers are uncountable.

However, you can only do something a finite numbers of times. That number might be unbounded, but it cannot be infinite. Mathematically, you can repeat a process an infinite number of times. But, you can only repeat your story a finite number of times. There is no way to produce an infinite number of stories.

If we are talking about something in the real world (or imagined about the real world), then we have a finite number of possibilities.

If we are talking about mathematics, then we can have a countable or uncountable infinite number of possibilities. However, if by a story we mean a sequence of words or a sequence of images (each of which is itself finite), then the number of possibilities is countable. It's possible mathematically to define an infinite story. Or, an infinite sequence of stories.

An example is to have the character toss a coin repeatedly. And the story is simply the sequence of heads and tails. Practically, that story or sequence must terminate at some point - even if there is no point at which it must terminate. Mathematically, that story or sequence can be assumed to continue indefinitely.

This is what I assume we are talking about here. In this case, if the coin has two sides (heads and tails), then it must land heads at some point in the story and tails at some point. If not, then the relevant probability is zero by definition.

In other words, if heads can happen (probability of heads not equal to zero), then it must happen.

My answer would be that everything that can happen must happen. In the sense that everything with a non-zero probability of happening in your story must happen at some point in your repeating story.

Note that the repeating part is important. For example, we could have a story where the character tosses a coin once to start. If it lands heads, then it lands heads thereafter and likewise if the first toss lands tails, then it lands tails thereafter. In this case, there is probability of 1/2 that the coin lands heads in your story, but unless that happens on the first toss, then it will never happen.

Finally, probability theory can be extended to the concept of a continuous probability distribution on an uncountable subset of the real numbers. However, real numbers and continuous probability distribtions are strictly mathematical things. And, trying to conceive of an uncountable number of stories becomes a tricky mathematical exercise. Not least because all but a countable subset of the real numbers are indescribable. I.e. most specific real numbers cannot be described or communicated in any way.

If you move into the strictly mathematical world of continuous probability distributions, then there is a sense in which everything has a probability of zero. This is effectively noting the the length of each real number is zero. The problem comes if you conclude things about actual probability distributions based on that observation. That's been discussed on here before and the Internet is full of people making dubious claims about probabilities in this respect.

Let me clarify some of the things about the situation this character is in.
The character is trapped in a different universe, a smaller one where he dies and dies again.
The implication we get from the 3 deaths shown in the show is that they have to be physically possible.

So he is turning into an elephant and gets magically blown up, might not be a possibility, but him dying from cancer is possible.

So, what does this mean In this context, will he eventually die from cancer? Are we 100% sure that he will die from cancer? Or turn into a pop star and live until 85 just to die and repeat the loops again?

These scenarios are physically possible, but I don't see how they are guaranteed to happen.

also, the time of death might make these scenarios uncountable, but time is quantized. Which i believe makes these deaths countable infinites.

This silly anime discussion might have more to it than meets the eye.

 

[PeroK]

So he is turning into an elephant and gets magically blown up, might not be a possibility, but him dying from cancer is possible.

Whether that is possible depends on the parameters of your story. If it's impossible by the rules of the story, then it can't happen no matter how often you repeat the story. If the parameters of the story allow that to happen, then you could calculate the probability in a single run of the story. And the probability of it happening in 3, 10 or $n$ runs of the story. And, if hypothetically, we run the story an unlimited number of times, then it must eventually happen.

So, what does this mean In this context, will he eventually die from cancer? Are we 100% sure that he will die from cancer? Or turn into a pop star and live until 85 just to die and repeat the loops again?

That all depends on how you construct your stories.

These scenarios are physically possible, but I don't see how they are guaranteed to happen.

They are only guaranteed to happen if, hypothetically or mathematically, you rerun your story an infinite number of times. And, they are actually possible within the parameters of your story.

also, the time of death might make these scenarios uncountable, but time is quantized. Which i believe makes these deaths countable infinites.

The continuous nature of time is not itself a factor. Events in a story are discrete. For example, there would have to be a first event, a second event and a third event etc.

Stories are not necessarily constrained by the laws of physics. If they were, there would be no "science fiction" genre.

 

[FactChecker]

However, you can only do something a finite numbers of times. That number might be unbounded, but it cannot be infinite. Mathematically, you can repeat a process an infinite number of times. But, you can only repeat your story a finite number of times. There is no way to produce an infinite number of stories.

That is true, but the question could be rephrased in another way. Instead of saying that it is done a countably infinite number of times, we can say that it is done until the result in question occurs. That is, the result in question is the stopping condition of the experiment. Then we ask if the probability of stopping is 1.

Then we have two distinct cases:
If the set of all possible results is finite (even if huge), the answer is that it will stop eventually.
If the set of all possible results is uncountably infinite (like randomly drawing a real number between 0 and 1 and asking if we will ever get exactly $\pi$), then the answer is that it will (probably) never stop.

 

[PeroK]

That is true, but the question could be rephrased in another way. Instead of saying that it is done a countably infinite number of times, we can say that it is done until the result in question occurs. That is, the result in question is the stopping condition of the experiment. Then we ask if the probability of stopping is 1.

Yes, and that naturally leads to the need for the limit of an infinite sequence. The probability of getting no heads after $n$ fair coin tosses is $\frac 1 {2^n}$. That is nonzero for any finite $n$, but the limit as $n \to \infty$ is zero.

No finite experiment can guarantee at least one head. But, we can use the mathematics of infinite sequences to define precisely what we mean by the event being inevitable.

 

[sairoof]

Stories are not necessarily constrained by the laws of physics. If they were, there would be no "science fiction" genre.

Possible withing the story's laws

Yes, and that naturally leads to the need for the limit of an infinite sequence. The probability of getting no heads after $n$ fair coin tosses is $\frac 1 {2^n}$. That is nonzero for any finite $n$, but the limit as $n \to \infty$ is zero.

No finite experiment can guarantee at least one head. But, we can use the mathematics of infinite sequences to define precisely what we mean by the event being inevitable.

If we repeat this experiment an infinite amount of times, wouldn't we guarantee that one of the sequences will have heads repeat infinitely?

 

[PeroK]

If we repeat this experiment an infinite amount of times, wouldn't we guarantee that one of the sequences will have heads repeat infinitely?

No. We will always get both heads and tails in every infinite sequence. If we have $N$ infinite sequences, the probability that one of the sequences is all heads is zero. And the limit as $N \to \infty$ is zero.

Rigorous mathematics was developed so that we can answer such questions precisely.

 

[jbriggs444]

Consider a drunkard's walk. We place the drunkard on an infinite grid. Once every second, he takes one step to a directly adjacent grid point.

In one dimension, this is a number line. There is a 100% probability that the drunkard will eventually return to his starting location. This is almost certain (probability 100%) to repeat infinitely many times.

In two dimensions, this is a plane of graph paper. The return to the starting position is again 100% probable.

In three dimensions, the return to the starting position is only 34% probable. Even though it is always possible for a path to proceed back to the origin, it is no longer almost certain.

 

[DaveC426913]

In this case, the circumstances of dying is what changes with each time.
So he might die from his skull crushed once by a car, and once by lifting heavy weights.

Yes. Review my post. I took it right down to the atomic level.

There are a very large number of configurations one's atoms can be jumbled into by a moving car, and a large number of configurations one's atoms can be jumbled into by a set of weights. And a host of other grisly tools of death. Together, they make a number even larger. But it is quite finite. if only simply because we are finite objects.

 

[jackjack2025]

I'd say, intuitively, that there are not an infinite number of discrete ways to die. It seems to me, that, with a finite structure such as a human body, with a finite number of atoms, it has to be so. While the number of ways those atoms could be separated from each other (or merely mixed together in non-functional ways) such that metabolism ceases may be very large, it is not infinitely large.

A lot of things come down to how you lay out the terms. In probability, you need a sample space, a sigma-algebra on that space, and a probability measure.

You framed the 'experiment' to do with atoms and what death might mean. If I say, every death in this anime show, the character is going to be wearing a t-shirt with a natural number on it that increases by one for each episode. Now there are countably infinite 'deaths', each distinct for me, because he has a different t-shirt on, regardless of the grisly method. Not distinct in your approach, because the t-shirt is irrelevant to your atoms idea.

Probability extends to countably infinite operations, but any discussion of uncountable infinite sets is outside the realm of measure theory.

To the original post, running an experiment an infinite number of times does not mean all possible outcomes will occur. If you have monkeys randomly hitting a typewriter for an infinite amount of times, you will get the complete works of Shakespeare, but that assumes certain things. Each key has a positive probability of being hit, the distribution of what keys are hit has an assumption, it assumes all the letters in Shakespeare's works are actually on this modern typewriter. If you are missing a key, then you will never get the works.

So is the original question asking, the Prob(A)>0 for some event A, and we re-run the experiment infinitely, will A occur with probability 1. That seems straightforward. Or is it asking Prob(A)=0, but still possible, will that happen if we run the experiment infinite times?

 

[DaveC426913]

umber on it that increases by one for each episode. Now there are countably infinite 'deaths',

An imaginative approach indeed!

 

[jackjack2025]

An imaginative approach indeed!

I guess I am trying to say, one should define what it is one is doing.

There is no such a thing (in reality) as tossing a coin or rolling a die infinitely many times. Nor do we have infinite atoms in our body as you point out. What there is, is some rigorous mathematics, from which you can draw results based on ones assumptions, and infinity in the context above means behaviour in the limit.

So as someone said, we can say if rolling a die infinitely, the Probability of always rolling 6 every time is 0. But it can still happen. But the statement is a bit 'loose'. What we mean is the probability of rolling a 6 every time, given a finite number N of rolls is modelled in a certain way, and the probability of rolling ten 6's in a row is higher than twenty 6's in a row (because we have set up our model in that way), then we can consider the behaviour of N in the limit towards infinity, which tends towards 0. But perhaps best not to think about infinity as something that is real, better to think of it as a concept for the limit.

I mention monkeys typing Shakespeare's works because there was a paper published not long ago which hit the news, saying that: the monkeys wouldn't be able to type all the works of Shakespeare, because it would take longer than the the life of the universe. So they sort of missed the point.

 

[Hornbein]

I once had a discussion with a layman on this topic. He said, "you might be right mathematically but I still don't believe it.". And that was that.

 

[WWGD]

I'm not clear on the layout of the problem. What prevents the exact same outcomes from repeating? Are you talking probabilities or how overall likely/probable events are? How do we determine if events are equal or not? Would being shot at, say, 2:01:00 p.m be different from being shot at 2:01:01, etc?

 

[DaveC426913]

I'm not clear on the layout of the problem. What prevents the exact same outcomes from repeating?

Nothing prevents it. Outcomes can repeat themselves.


Or are you alluding to what the randomizing factor is? i.e. why it isn't the same outcome every time?

I mean, something has to drive different outcomes. A flat sheet of paper has only two possible final configurations (OK, maybe a little more), but it can't have an infinite number of possible final configurations.

Is this what you're driving at?

I think - judging by the opening post - the randomizing element in the scenario is the environment. Every passing vehicle, swinging I-beam, suspended grand piano and incoming meteor strike.

 

[WWGD]

Yes, I guess what the randomizing factor is.