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Infinite coin flip probability


Lord Antares
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...however, infinity complicates the issue.

 

Doesn't it just.

 

However, on the infinite ruler example, the probability of getting to zero changes based on previous tosses. So, if you flipped heads 699 times in a row, you would be at 700, making it less likely that you would eventually return to zero. This is all fine and dandy, however, infinity complicates the issue. I am not sure if the probability is 1 (the opposite of the previous example's 0), or if it is somehow changed based on the fact that previous tosses impact the result. It makes sense to me that it would be, but it also makes sense that it wouldn't. I am not certain.

 

OK, try this. Say we have done some flips and find ourselves on step 700 (or any number). We can then ask the question if we flip the coin infinitely many more times, what is the probability of getting back to step 0. That is, what is the probability of getting a sub-sequence within our infinite sequence such that it takes us back to step 0?

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Well, that is what I'm asking you.

But look at this. Take two different examples

 

1) We are on meter 1 and we must find the probability that we return to 0 with infinite flips

2) We are on meter 700 and we must find the probability that we return to 0 with infinite flips

 

Although mathematically, we might find that the probability of both instances is 1, isn't it technically more likely that instance 1 will bring us back to 0, given the almost surely clause?

 

I draw a parallel to wtf's earlier example:

 

Let us say that we must pick one number in the set of natural numbers (or any set), the probability that we guess which one gets picked randomly is 0, since we have a 1/infinity chance of guessing it right. However, let us say that we must pick two numbers and we must guess one of them. The probability would be 2/infinity. Although the probability in both instances is 0, is the latter, or is it not, technically more probable? This may not make sense, but in my opinion, it is a very interesting question.

 

By the same logic, if the must pick a number in the set of real numbers, rather than naturals, does the probability decrease, even though it is still 0? I think that the answer to this has bearing on our infinite ruler example.

 

This is a very important question to me, answers are most appreciated.

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Well, that is what I'm asking you.

But look at this. Take two different examples

 

1) We are on meter 1 and we must find the probability that we return to 0 with infinite flips

2) We are on meter 700 and we must find the probability that we return to 0 with infinite flips

 

Although mathematically, we might find that the probability of both instances is 1, isn't it technically more likely that instance 1 will bring us back to 0, given the almost surely clause?

 

The probability is one in both cases. If you ask the question which sequence do we expect to reach zero first then we would say the one that starts on one. But we don't care how long it takes either to reach zero, only that at some point in the infinite flips they do reach zero. And they do both reach zero, at some point, with probability one.

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Let us say you have a ruler laid out on the floor, with the spacings arranged such that 1 increment = 1 meter. You are standing at 0.

You start flipping a coin an infinite number of times. When you get heads, you move 1 meter forwards (positively) on the ruler. When you get tails, you move 1 meter backwards. If you flip tails at 0, you stay at zero. Let's say that you flipped the coin once and it landed on heads, so you are now standing at 1 meter. With a potentially infinite number of coin tosses, what are the odds of eventually arriving at 0? How different would it be if the ruler was infinite as opposed to finite?

 

The first question is

 

Is there a sequence of flips that will lead one back to zero?

 

If so that sequence must have a probability.

 

Second question is there a sequence with probability = 1 that will lead you back to zero?

 

If so then there are no other sequences available.

 

Third question are there other sequences available?

 

If yes then there is no sequence that will lead you back to zero, with probability 1.

 

In terms of the coin flips there is one that is all heads and so it will carry you further and further away from zero.

 

This sequence must have a probability (however low) by hypothesis.

 

Here is where we must distinguish between finite and infinite flips.

 

For finite flips we are done. We have found at least one sequence that terminates without reaching zero.

 

For infinite flips we need to confirm that there are more than one infinite sequences and at least one of them never reaches zero.

 

Well all heads, if carried on without termination will never reach zero.

 

As will two heads followed by one tail (or 10 tails) followed by all heads again.

 

So we are done as well for infinite sequences.

 

 

Incidentally this shows an example of the fact that it does not matter how many finite terms you add at the beginning of an infinite sequence, it does not affect the eventual convergence (or not) of that sequence.

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Well, that is what I'm asking you.

But look at this. Take two different examples

 

1) We are on meter 1 and we must find the probability that we return to 0 with infinite flips

2) We are on meter 700 and we must find the probability that we return to 0 with infinite flips

 

Although mathematically, we might find that the probability of both instances is 1, isn't it technically more likely that instance 1 will bring us back to 0, given the almost surely clause?

Not when you give me infinite flips. With infinite flips, a symmetric random walk on 1-D will visit every point. It doesn't matter where you start. Given infinite flips, it will visit every point.

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Not when you give me infinite flips. With infinite flips, a symmetric random walk on 1-D will visit every point. It doesn't matter where you start. Given infinite flips, it will visit every point.

I just skimmed through the topic of one-dimensional random walks on the integers over at Wiki and although I don't have any more insight into any of this, I did discover a whole lot more things I'll never know anything about.

 

https://en.wikipedia.org/wiki/Random_walk#One-dimensional_random_walk

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Just a note on random walks.

 

 

Not when you give me infinite flips. With infinite flips, a symmetric random walk on 1-D will visit every point. It doesn't matter where you start. Given infinite flips, it will visit every point.

 

Yes indeed a good answer to the quote, but this is not a standard random walk.

 

Random walks don't terminate.

That is all sequences in a random walk are infinite, as you note and as does the Wiki article wtf refers to.

 

This question has a termination clause in its contract.

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The probability is one in both cases. If you ask the question which sequence do we expect to reach zero first then we would say the one that starts on one. But we don't care how long it takes either to reach zero, only that at some point in the infinite flips they do reach zero. And they do both reach zero, at some point, with probability one.

 

 

Not when you give me infinite flips. With infinite flips, a symmetric random walk on 1-D will visit every point. It doesn't matter where you start. Given infinite flips, it will visit every point.

 

I understand this. I understand that, given infinite time, everything must happen with probability 1. HOWEVER, I am trying to be 100% technical here. Since getting to 0 from either 1 or 700 with infinite flips yields the probability of 1, it should happen, but technically, it doesn't have to. Technically, it is possible to flip only heads forever. To assume otherwise would be a gabmler's fallacy. I understand that this would never happen in a real example, I'm just trying to be technical.

 

Another example is:

 

1) A number from all the natural number is randomly drawn and you need to pick one and guess it.

2) You need to pick 2 numbers and guess one of them.

 

Is the probability of both technically the same? The probability of guessing either is 0, but shouldn't it be a technically higher probability for the second case? Even though both are 0% of infinity and so, in a way, the same. Also, why not infinitesimal chance, rather than 0? I would think that would be more apt. At least it woul be clearer in my view. Of course, mathematics is complicated and has been developed for a long time, so there must be precise reasons.

 

EDIT: I think studiot addressed this.

 

 

The first question is

 

Is there a sequence of flips that will lead one back to zero?

 

If so that sequence must have a probability.

 

Second question is there a sequence with probability = 1 that will lead you back to zero?

 

If so then there are no other sequences available.

 

Third question are there other sequences available?

 

If yes then there is no sequence that will lead you back to zero, with probability 1.

 

 

OK, I follow.

 

 

 

In terms of the coin flips there is one that is all heads and so it will carry you further and further away from zero.

 

 

Not only one that is all heads, but there are an infinite number (in the inifnite case) of ones that are MORE heads than tails, achieving the same thing!

 

 

 

This sequence must have a probability (however low) by hypothesis.

 

 

Yes.

 

 

 

For infinite flips we need to confirm that there are more than one infinite sequences and at least one of them never reaches zero.

 

Well all heads, if carried on without termination will never reach zero.

 

As will two heads followed by one tail (or 10 tails) followed by all heads again.

 

So we are done as well for infinite sequences.

 

 

Incidentally this shows an example of the fact that it does not matter how many finite terms you add at the beginning of an infinite sequence, it does not affect the eventual convergence (or not) of that sequence.

 

I'm not sure I'm understanding you correctly here. I agree with everything you said here but I'm confused as to the conclusion of this. You say that an all heads, as well as MORE heads than tails infinite sequence will never lead us back to zero, which I agree with. But I'm not sure what the conclusion means.

 

Unless...hmm maybe this:

Is it that we cannot prove that there are more infinite sequences while on meter 700 than on meter 1 which lead back to zero, which in turn means that there are an infinite amount of sequences for both, hence equal probability?

Ahhh, if so, it makes sense to me now. Do reply back to say if this is what you meant.

Edited by Lord Antares
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The reasoning is inherent in the way you have set up the problem.

 

You have a termination criterion viz a sequence terminates when 0 is reached, otherwise the process continues indefinitely.

 

Thus all finite sequences terminate. (on 0)

 

and all terminating sequences are finite.

 

By similar reasoning an infinite sequence has not reached 0 and continues indefinitely without reaching 0

 

But you asked for the probability of reaching zero.

 

I have been trying to offer more than just the answer to this question, but let's get the answer out of the way.

 

So we must consider finite sequences and in particular how many finte sequences are there?

 

Well consider this:

 

In order to reach 0 a process must first reach 1 and then flip a tail on the next throw.

 

The sequence to reach 1 must be finite as just noted above.

 

But we can extend any finite sequence at all that goes ........WXYT, where at Y the sequence is at 1 on the ruler, by simply adding an H between the Y and the T thus

 

WXYHT

 

Which means the sequence is now at 1 on the ruler again after one more step

 

So our new extended sequence is of the same form as before ie a single T off a termination to 0 on the ruler.

 

This proceedure can be repeated indefinitely, each time extending the length of the sequence by one step.

 

Since WXY is completely general, every finite sequence can be extended this way indefinitely.

 

Which leads to conclude that the number of finite sequences can be considered infinite.

 

Which leads to the conclusion that each one has zero probability.

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Ahaaa, I think I understand.

The only way that a sequence will get terminated is if it reaches zero. Therefore, all finite sequences must end up at zero. Therefore, the probability that a given finite sequence will lead us to zero is 1. If a sequence veers a way from zero constantly, it will never be a finite sequence, unless it reaches zero.
Furthermore, there is an infinite amount of sequences which lead to zero. Therefore, any given sequence has a probability 0 of occuring. But funnily enough, one of the sequences must happen, and therefore, a probability zero event must happen! Does this sum it up?
Nice post, +1.
However, there is also an infinite number of infinite sequences. Any infinite sequence CANNOT lead back to zero, as it would terminate there and be finite. So, technically, it is possible that a sequence such as HTHTHT... continues indefinitely, which will be an infinite sequence and never terminate. This sequence also has a probability 0 of happening, but it is technically posssible.
Edited by Lord Antares
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  • 1 month later...

Can you give an example of a probability problem involving infinity that gives two correct and inconsistent answers?

one example I saw a Numberfile youtube video where the infinite series of something like

1+2+3+4... infinity = -1/12.

 

I could be remembering it wrong and it could have been the product instead of the sum, but the point is that the series logically expands to infinity but there was a proof that used a combination of other infinite series to find the -1/12 answer to be true.

 

I tried to search for that video, but couldn't find it.

 

Another simpler example I heard when I was very young.

 

If you lived in a world with an infinite population of immortals where one immortal was born in some random location every year and everyone knew their own age, then from the point of view of any random immortal you happened to pick, what are the odds of him running into someone younger than he was.

 

The answer is infinitessimally small because there are an infinite number of older people while theres only a finite number younger so it doesnt even matter how old he was, the odds against meetig someone younger is still infinitessimally small.

 

And yet from the perspective of a third party, one person is always younger than another person when they meet.

 

Before thinking about it too much, it could be like saying that if infinity could exist in the universe, then conservation laws would probably not be true. At least not in the form that they are currently understood.

Edited by TakenItSeriously
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I think that first sum is related to the Riemann Hypothesis, or at least the zeta function, and i believe uses a different summation method - i don't know nearly enough to comment further though.

 

 

The second case is just a manifestation of it being possible that probability zero events can occur. This is uncontroversial, if somewhat clumsy language.

 

 

I'd go further and say that without infinity, particularly the idea of limits, statistics would be unable to progress beyond it's state several centuries ago. The Law of Large Numbers and the Central Limit Theorem are absolutely fundamental to probability theory and both rely on asymptotic results. Are you saying these theorems are incorrect and we should throw away the last few centuries of statistical knowledge reliant on them?

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I think that first sum is related to the Riemann Hypothesis, or at least the zeta function, and i believe uses a different summation method - i don't know nearly enough to comment further though.

 

 

The second case is just a manifestation of it being possible that probability zero events can occur. This is uncontroversial, if somewhat clumsy language.

 

 

I'd go further and say that without infinity, particularly the idea of limits, statistics would be unable to progress beyond it's state several centuries ago. The Law of Large Numbers and the Central Limit Theorem are absolutely fundamental to probability theory and both rely on asymptotic results. Are you saying these theorems are incorrect and we should throw away the last few centuries of statistical knowledge reliant on them?

I should have clarified that I don't believe in physical manifestations of infinity such as the type given in the OP.

 

Conceptually, however, I agree. I think infinity is a very important concept as a boundary condition. IMHO, I tend to think of approaching infinity without ever reaching infintity for purposes of asymptotes or approximations, at least in terms of anything physical in the domain of the observable Universe, as the proper way to treat infinity.

 

My only exception being, when falling through the Event Horizon of a Black Hole at the speed of light where infinity goes crazy but in a self consistent, non-singularity kind of way, which is a whole other topic. That I can't talk about and probably won't be accepted for decades.

 

I seriously doubt that if the world can wait that long, before the current mistakes of humanity become irreversible.

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I should have clarified that I don't believe in physical manifestations of infinity such as the type given in the OP.

 

I never gave my belief on the physical manifestation of inifinity. Probability is very interesting to me. I am interested the most in its technical aspects; its applicability may or may not interest me, depending on the situation. Obviously, the OP was not a real-life situation. It was a curiosity to me to see how mathematics handles infinity in probablity. You can learn from it regardless of the fact that it may not ever be applicable.

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I think that first sum is related to the Riemann Hypothesis, or at least the zeta function, and i believe uses a different summation method - i don't know nearly enough to comment further though.

 

 

The second case is just a manifestation of it being possible that probability zero events can occur. This is uncontroversial, if somewhat clumsy language.

 

 

I'd go further and say that without infinity, particularly the idea of limits, statistics would be unable to progress beyond it's state several centuries ago. The Law of Large Numbers and the Central Limit Theorem are absolutely fundamental to probability theory and both rely on asymptotic results. Are you saying these theorems are incorrect and we should throw away the last few centuries of statistical knowledge reliant on them?

 

It is the Ramanujan Summation - which gives an answer to the value of an infinite divergent series; in normal use there is no sum to an infinite divergent series - this is a special version which bears passing resemblance to the sum and allows further study of the concept. You are also correct in that it is used in zeta function methodologies

 

I have read that Ramanujan summation is also used in normalization of certain quantum field theories - which in turn give positive predictions in terms of real world results. So it is weird and a bit unwholesome but there is definitely very important maths there

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I have read that Ramanujan summation is also used in normalization of certain quantum field theories - which in turn give positive predictions in terms of real world results. So it is weird and a bit unwholesome but there is definitely very important maths there

In the video, they meantioned that the series was was used in 26 dimensional String Theory as well, which is why I speculated that Infinity could be realized in reality but requires extra dimensions.

 

FWIW, certain aspects of infinity are included in my own interpretation of GR which involves falling through the EH of BHs at the SoL.

Edited by TakenItSeriously
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In the video, they meantioned that the series was was used in 26 dimensional String Theory as well, which is why I speculated that Infinity could be realized in reality but requires extra dimensions.

 

FWIW, certain aspects of infinity are included in my own interpretation of GR which involves falling through the EH of BHs at the SoL.

 

QFT is predictive and has been tested to phenomenal precision. String Theory has not once been tested to any level of prediction; basically almost all complex maths will find itself bound up in string theory in some way.

 

Whether string theory turns out to be empirically predictive or theoretically useful in physics, one thing is for certain - it is mathematics and modelling of the highest possible order; perhaps mankind's greatest abstract achievement.

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