A billion consective zeros in pi

[Thecla]

Pi is an irrational number with an unpredictable distribution of the numbers 0 to 9. Yet patterns do emerge. Can we say with certainty that there is some point in pi where there are a billion consecutive zeros? Is there any way to calculate the number of digits of pi that must be expanded to have a 50% probability of this occurring?
Is this analagous to the statement that if you flip a coin long enough you will with certainty get one billion heads in a row ?

 

[CRGreathouse]

Can we say with certainty that there is some point in pi where there are a billion consecutive zeros?

This has not been proven. It would follow from pi's normality, if that was proven.

Is there any way to calculate the number of digits of pi that must be expanded to have a 50% probability of this occurring?

Sure. As a very rough first guess, it will be somewhere around 10^1000000000.

 

[DrRocket]

Pi is an irrational number with an unpredictable distribution of the numbers 0 to 9. Yet patterns do emerge. Can we say with certainty that there is some point in pi where there are a billion consecutive zeros? Is there any way to calculate the number of digits of pi that must be expanded to have a 50% probability of this occurring?
Is this analagous to the statement that if you flip a coin long enough you will with certainty get one billion heads in a row ?

No.

The digits of pi are not random -- they are the digits of pi. That completely determines them.

Probability theory does not apply.

 

[Office_Shredder]

No.

The digits of pi are not random -- they are the digits of pi. That completely determines them.

Probability theory does not apply.

But we don't know what the digits of pi are, so asking what the probability of them being is a fair question.

Not to go off track, but if I flip 10 coins, then tell you that at least five landed on heads, the question of what is the probability that all the coins landed on heads is a valid one, even though all the coins have been flipped and are completely determined at this point.

Back to the question at hand: it's unknown what statistical pattern the digits of pi may have. It's conjectured that essentially the digits are independent and uniform; i.e. 1/10 of the digits are 1, 1/10 are 2, etc. and that other strings of digits are also equally likely: 1/100 of all two digit strings are 01, 1/100 of all two digit strings are 23, etc. This is called normality. Unfortunately nobody knows how to prove whether a number is normal or not (well, you can prove some numbers are not normal pretty easily). If pi is normal, then by definition a string of a billion 0's must occur

 

[DrRocket]

But we don't know what the digits of pi are, so asking what the probability of them being is a fair question.

No, it is most certainly not a fair question, or even a meaningful question.

Probability is one of most misunderstood and misapplied branches of mathematics. This is a a good example.

The digits of pi have nothing to do with any probability measure and therefore asking a probabilistic question is utterly meaningless.

It is quite true that the digits of pi are sometimes used to general a sequence of numbers that are called pseudo-random because of the lack of "predictability" of the numbers, bu the operative term is the prefix psuedo. They are not really random in any meaningful sense.

Not to go off track, but if I flip 10 coins, then tell you that at least five landed on heads, the question of what is the probability that all the coins landed on heads is a valid one, even though all the coins have been flipped and are completely determined at this point.

This example has nothing whatever to do with the original question. In this case you have what one ASSUMES to be a random process and that one knows that process to have produced at least 5 heads. You then ask the conditional probabiliy, given that knowledge, that all the coins were heads. In fact what you are asking is the relative frequency with which all heads will turn up in a fair flip of a 10 coins given that at least 5 heads are observed to occur.

If you are asking the probabiliy that all heads have turned up in one specific trial, then that is not a question of probability, and you are again misapplying the theory, or at least re-interpreting the question as outlined above so that probability theory does actually apply. I realize that what you have posed is a fairly typical textbook question, but such a question is really just shorthand for what I outlined above. One needs to be very careful in applying probability theory, or you will fool yourself.

Back to the question at hand: it's unknown what statistical pattern the digits of pi may have. It's conjectured that essentially the digits are independent and uniform; i.e. 1/10 of the digits are 1, 1/10 are 2, etc. and that other strings of digits are also equally likely: 1/100 of all two digit strings are 01, 1/100 of all two digit strings are 23, etc. This is called normality. Unfortunately nobody knows how to prove whether a number is normal or not (well, you can prove some numbers are not normal pretty easily). If pi is normal, then by definition a string of a billion 0's must occur

If you will look closely at the definition of a normal number, you will find that the definition is in terms of a limit that intuitively seems similar to the law of large numbers, but in reality that the definition does not actually invoke probability theory at all. The layman's definition in terms of a (probability) distribution of the digits is misleading in that regard.

I turns out that relatively little is actually known about normal numbers. It is not, for instance, known if pi is normal. In any case the digits of pi are not random and you cannot appeal to probability theory to make determinations about them.

 

[jackmell]

This has not been proven. It would follow from pi's normality, if that was proven.
Sure. As a very rough first guess, it will be somewhere around 10^1000000000.

I did a little empirical work with that but with e. Where is the smallest 10-digit prime in e? Well, it turned out to be within the range of where we would have expected it to occur if we assumed the digits were random: Right around 1.9 billion.

As a side-note, I think the location of large numbers in the decimal digits of any irrational number might be the basis of a new encryption algorithm as even the 10-digit prime calculation was quite CPU intensive although I'm not sure it would be resistant to just random guesses.

 

[CRGreathouse]

As a side-note, I think the location of large numbers in the decimal digits of any irrational number might be the basis of a new encryption algorithm

Bad idea.

 

[Coto]

This thread peaked my interest because of DrRocket's reply. I was wondering if you might go into more depth about the fallacy of applying probability theory to the digits of pi?

It is quite true that the digits of pi are sometimes used to general a sequence of numbers that are called pseudo-random because of the lack of "predictability" of the numbers, bu the operative term is the prefix psuedo. They are not really random in any meaningful sense.

In the pseudo-random sense, can we still not say something about statistical distributions of the numbers? Wouldn't this be the surest way to test whether something is pseudo-random? If this is the case, why can't we say something about the probability of pi attaining the same digit multiple times?

This example has nothing whatever to do with the original question. In this case you have what one ASSUMES to be a random process and that one knows that process to have produced at least 5 heads. You then ask the conditional probabiliy, given that knowledge, that all the coins were heads. In fact what you are asking is the relative frequency with which all heads will turn up in a fair flip of a 10 coins given that at least 5 heads are observed to occur.

If we make the assumption that pi is a normal number can we say something about whether multiple instances of a digit will appear? Is there a correct way to apply a probability to this? If not, perhaps you could give an analogy to clear up why it is not possible.

Thanks,
Coto

 

[disregardthat]

Bad idea.

Why is this a bad idea? I'm just curious.

If this is the case, why can't we say something about the probability of pi attaining the same digit multiple times?

Say any well-defined statement about the decimal expansion of pi, and the status of it would be either true or false. The probability of 777 being attained in the first 10,000 digits is either 1 or 0, and the number of occurrences is a unique natural number, not a random variable.

 

[DrRocket]

This thread peaked my interest because of DrRocket's reply. I was wondering if you might go into more depth about the fallacy of applying probability theory to the digits of pi?



In the pseudo-random sense, can we still not say something about statistical distributions of the numbers? Wouldn't this be the surest way to test whether something is pseudo-random? If this is the case, why can't we say something about the probability of pi attaining the same digit multiple times?



If we make the assumption that pi is a normal number can we say something about whether multiple instances of a digit will appear? Is there a correct way to apply a probability to this? If not, perhaps you could give an analogy to clear up why it is not possible.

Thanks,
Coto

The problem with testing for "pseudo-randomness" is that there is no definition for "pseudo-random" and hence no test. Pseudo-random is a term that means roughly "we would like to call it random, but we have no definition for random, so we'll use pseudo-random instead".

Here is the basic problem. The term random comes up in probability theory. Probability theory starts with the hypothesis that one has a set, a sigma algebra of subsets, and a positive measure defined on the sigma algebra that measures the entire space as 1 -- thgis entire thing being called a probability space. A random variable is then just a measureable function on the probability space. Thus, following this formulation of Kolmogorove, probabiltiy theory is just a branch of the general theory of measure and integration.

So, probability theory starts with a probability space. It assumes that you mave the probability measure defined, and it annoints certain functins as "random" by virtue of their measurability (in the sense of abstract measure theory, which has nothing to do with being measurable in any physical sense using any sort of intstruments).

Pseudo-random is even more loosely defined. It is essentially meaningless mathematically and means generally "unpredictable" in some not-very-well-defined sense. A fairly typical pseudo-random generator starts calulating the digits of pi and assumes that that generates a sample path for a random variable of integers uniformly distributed on the interval [0.9]. There is no theoretical basis for this, but it more or less works in practice (since there is no way to actually check it, there is no way to reallyt dispute it either). What you get, which it looks "random", whatever that means, it also completely deterministic, and the algorithm will generate exactly the same "random sequence" of integers every time you run the algorithm.

Probability theory has some fairly deep theorems, notably the central limit theorem, and the law of large numbers. But to apply those theorems you need to first fulfill the hypotheses, and that requires that one have a probability space. The digits of pi do nor provide any sort of probability space on which to apply the basic theory. Thus you cannot reach conclusions by applying theorems the hypotheses for which are unfulfilled. It is just that simple.

There are no theorems that apply to pseudo-random variables, simply because there is no useful definition for the critters. You have something that is not quite a random variable so the best that you can do is apply something that is not quite a theorem -- in short you are whistling in the dark.

Now, you might be able to formulate some sort of probability question that intuitively relates to some similar question about pi. But that will not directly answer the questions one one like to answer about pi specifically.

Sometimes a suggestive probabilistic problem is formulated and answered along the road to answering the question that everyone would like to have answered. That is sometimes useful and sometimes not. Such things have been tried with the Riemann Hypothesis for instance, but don't receive a lot of press because they really don't answer the basic question.

The problem is that just because something is not known does not imply that one can apply the methods of probability to answering it. Some questions are simply not probabilistic in nature. It may be that 10% of the people in the U.S. are named "Bob" but that does not mean that there is a 10% probability that your name is Bob just because I don't know your name. Your name is either Bob or it is not. There is nothing random about it. What is true is that if I select a person at random from the population then I can expect to find a Bob about 1 time in 10 (on average), but that has nothing whatever to do with what your specific name is.

Similarly the digits of pi are whatever the digits of pi are and probability has nothing whatever to do with it.

A normal number has certain properties, by definition, concerning the occurence of digits as the number of digits becomes large. So, if you assume that pi is normal then, by assumption, the conclusions that would apply to any normal number will apply to pi. The rub is that you have no idea if your assumption is correct or not, and hence no idea whether the conclusions are valid or not.

 

[Coto]

Thanks DrRocket. I have had the luxury of taking an introduction to measure theory (as seen through the eyes of Lebesgue and Haar measure) but I have never seen it in its application to probability theory.

The problem is that just because something is not known does not imply that one can apply the methods of probability to answering it. Some questions are simply not probabilistic in nature. It may be that 10% of the people in the U.S. are named "Bob" but that does not mean that there is a 10% probability that your name is Bob just because I don't know your name. Your name is either Bob or it is not. There is nothing random about it. What is true is that if I select a person at random from the population then I can expect to find a Bob about 1 time in 10 (on average), but that has nothing whatever to do with what your specific name is.

Similarly the digits of pi are whatever the digits of pi are and probability has nothing whatever to do with it.

Are you saying something like this: If we take 100 people, name 10 of them Bob and the rest as unique names, then we have 91 names total. If you encounter one of these 100 people then there's actually 1/91 chance that their name is Bob. However our likelihood of encountering a Bob is 1/10?

It seems to me that the English language can be ambiguous when talking about probability theory and my example may well be an artifact of this ambiguity.

 

[DrRocket]

Are you saying something like this: If we take 100 people, name 10 of them Bob and the rest as unique names, then we have 91 names total. If you encounter one of these 100 people then there's actually 1/91 chance that their name is Bob. However our likelihood of encountering a Bob is 1/10?

No.

I am saying that probability theory can tell you something about the expected frequency of encountering a particular sample path, or some class of sample paths. But it cannot tell you anything at all about the attributes of some specific sample path.

So, if 10% of the people are named Bob then if you pick someone at random and ask their name, then repeat this a large number of times you find a "Bob" about one time in ten. But that has nothing do with the probability that your personal name is Bob. You know your name, and either it is Bob or it is not Bob and probability has nothing to do with that.

So think about this in the context of a decimal expression for pi. There are infinitely many possible decimal representations for real numbers (uncountably many in fact) and if try to construct some sort of probability measure with the assumption that the likelyhood of any particular digit in the nth position is the same as any other digit, you can do that. But a couple of things then happen.

First, you will find that you cannot assign a probability to any infinite decimal expansion, but rather the fundamental sets that generate your sigma algebra will be of the form, some specific set of digits in the first n places, and everything beyond n arbitrary. So you can sensibly talk about the probability of some pattern in the first n digits, for a fixed but arbitrary n, but you cannot talk about the probability of some pattern of digits in any specific full (infinite) expansion, and in particular not about pi.

Second you will find that the probability associated with any actual infinite decimal expansion is zero. That is to be expected, since there are infinitely many of them, all "equally likely" and the only that can happen is if the associated measure/probability is zero. So in this case you are talking about something on a set of measure zero, which is not going to lead anywhere.

So, the bottom line is that probability theory cannot tell you anything about pi, or about any specific number.

Asking whether pi is normal or not does not affect his. Normality is not really related to probability per se, but seems to do so in terms of the definitiion of the term which considers asymptotic "distributions" of the density of the digits. In fact all that one has done by introducing normality is to give a name to the problem.

 

[Coto]

I am saying that probability theory can tell you something about the expected frequency of encountering a particular sample path, or some class of sample paths. But it cannot tell you anything at all about the attributes of some specific sample path.

So, if 10% of the people are named Bob then if you pick someone at random and ask their name, then repeat this a large number of times you find a "Bob" about one time in ten. But that has nothing do with the probability that your personal name is Bob. You know your name, and either it is Bob or it is not Bob and probability has nothing to do with that.

Interesting (please bear with me), but can you not say something about the probability of guessing the right name given that you have a finite number of names to choose from (as per example above)?

First, you will find that you cannot assign a probability to any infinite decimal expansion, but rather the fundamental sets that generate your sigma algebra will be of the form, some specific set of digits in the first n places, and everything beyond n arbitrary. So you can sensibly talk about the probability of some pattern in the first n digits, for a fixed but arbitrary n, but you cannot talk about the probability of some pattern of digits in any specific full (infinite) expansion, and in particular not about pi.

So assuming we truncate pi after a very large n can we then say something about encountering a sequence of digits? Would the statement of normality of the n digits help in any way?

So, the bottom line is that probability theory cannot tell you anything about pi, or about any specific number.

So long as that number is real? Take for example that you look at all numbers that begin with 3 and are truncated to 10 decimal places. Then can you say something about encountering a sequence of digits? Is this the same question as whether or not a specific number (say pi truncated to 10 digits) has that sequence of digits, assuming you don't know what pi is aside from the fact that it starts with a 3?

(I realize that the last question is a revolving door of sorts and it stems from my confusion about its answer. The first two questions may very well speak for the third.)

 

[DaveC426913]

OK, so getting back to the OP, the question is valid, if not satsifactorily formed.

- pi has an infinite number of digits
- we can, so far, discern no pattern to the digits

We should be able to state that a sequence of one billion zeros will occur somwehere in its length. How do we state that without using probability?

 

[Office_Shredder]

We should be able to state that a sequence of one billion zeros will occur somwehere in its length. How do we state that without using probability?

Even if the digits of pi were chosen randomly, you can't state that it will occur, because that might not be true. For example, the if I pick digits completely at random, I might get the sequence 0,0,0,0,0,0,0,... It will never actually happen of course, but there is nothing that precludes the possibility.

What you can do is state that it will occur with probability one. It turns out that once you start having a large enough sample space, things that have a probability of zero can actually occur (for example, the string of 0's above).

DrRocket is getting hung up on this idea that pi's digits are not random, but in practice many useful conjectures can be made by assuming that things that are fixed have random distributions. This is not uncommon in number theory; for example when looking at the Goldbach Conjecture, you can start by assuming a probabilistic distribution of the primes:
http://en.wikipedia.org/wiki/Goldbach's_conjecture#Heuristic_justification

This isn't a proof but is a strong method of developing conjectures and a sense for whether they should be true. And if they end up not being true, it may mean something really weird is going on that should be looked into further.

 

[DrRocket]

Even if the digits of pi were chosen randomly, you can't state that it will occur, because that might not be true. For example, the if I pick digits completely at random, I might get the sequence 0,0,0,0,0,0,0,... It will never actually happen of course, but there is nothing that precludes the possibility.

What you can do is state that it will occur with probability one. It turns out that once you start having a large enough sample space, things that have a probability of zero can actually occur (for example, the string of 0's above).

DrRocket is getting hung up on this idea that pi's digits are not random, but in practice many useful conjectures can be made by assuming that things that are fixed have random distributions. This is not uncommon in number theory; for example when looking at the Goldbach Conjecture, you can start by assuming a probabilistic distribution of the primes:
http://en.wikipedia.org/wiki/Goldbach's_conjecture#Heuristic_justification

This isn't a proof but is a strong method of developing conjectures and a sense for whether they should be true. And if they end up not being true, it may mean something really weird is going on that should be looked into further.

I am not hung up on anything.

Your example of a probabilistic version of the Goldbach Conjecture is quite in keeping with the probabilistic version of the Riemann Hypothesis that I mentioned earlier. Neither is a path to a proof, and both are suggestive, which is their role.

You cannot assume that the digits of pi are random. They are not -- they are the digits of pi. You can study what sort of strings are likely to occur and with what frequency, in a random process involving selection of digits between 0 and 9, but there is no way to connect that to the decimal expansion of pi. It is as simple as that.

 

[DrRocket]

OK, so getting back to the OP, the question is valid, if not satsifactorily formed.

- pi has an infinite number of digits
- we can, so far, discern no pattern to the digits

We should be able to state that a sequence of one billion zeros will occur somwehere in its length. How do we state that without using probability?

You can state anything that you like -- all it takes is a pen.

The question is what you can prove. And there is nothing that you can prove in this regard using the theory of probability. It simply does not apply.

The fact that you can discern no pattern to the digits is simply an admission of ignorance. It is not an invitation to use some inapplicable technique.

 

[Office_Shredder]

You cannot assume that the digits of pi are random. They are not -- they are the digits of pi. You can study what sort of strings are likely to occur and with what frequency, in a random process involving selection of digits between 0 and 9, but there is no way to connect that to the decimal expansion of pi. It is as simple as that.

The distribution of prime numbers is not random either. What's your point? I never said you get a proof out of any such argument, just that it is a good way to develop conjectures probabilistically.

 

[DaveC426913]

I see Dr. Rocket's point.

pi is a fixed number in the universe. That fixed number might very well be 3.141592 ... 2323232323232 ad infinitum. There is no reason to think that any given sequence of numbers must occur within it, even if it is infinite.

 

[Office_Shredder]

pi is a fixed number in the universe. That fixed number might very well be 3.141592 ... 2323232323232 ad infinitum. There is no reason to think that any given sequence of numbers must occur within it, even if it is infinite.

Did I not just say essentially that with my 0,0,0,0... example?

 

[DaveC426913]

Did I not just say essentially that with my 0,0,0,0... example?

Frankly, I didn't see what you were getting at, no. You seemed to be using the same flawed idea of pi being random by comparing pi to a random sequence of digits.

 

[DrRocket]

I see Dr. Rocket's point.

pi is a fixed number in the universe. That fixed number might very well be 3.141592 ... 2323232323232 ad infinitum. There is no reason to think that any given sequence of numbers must occur within it, even if it is infinite.

Right, you have the basic idea.

BTW pi cannot actually be 3.141592 ... 2323232323232 ad infinitum because the repeating pattern results in a rational number, and we know that pi is not rational.

 

[DrRocket]

The distribution of prime numbers is not random either. What's your point? I never said you get a proof out of any such argument, just that it is a good way to develop conjectures probabilistically.

My point is mathematics.

The distribution of prime numbers, and the prime number theorem that describes it roughly, have nothing to do with probability.

 

[DaveC426913]

Let's look at a simpler example: The number of centimetres in an inch.

So far, our technology has only been able to calculate that number to 2 sig digs: i.e. there are 2.5cm in an inch.

What are the chances that a 9 will appear in the number? The chances that the 3rd number of the sequence will be a 9 are ... what? ... 1 in 10?


Nope. No matter how many times we discover the rest of the sequence, the probability of the third number being 9 is zero.

Why? Because the third number is 4. And it will always be 4 until the end of time. There is no probability involved.

 

[DaveC426913]

BTW pi cannot actually be 3.141592 ... 2323232323232 ad infinitum because the repeating pattern results in a rational number, and we know that pi is not rational.

3.1415926 ...2323234232323

Better?

Hm. No. Now I'm having doubts again.

No matter how many times I throw a number 4 into the mix of 23s, eventually the pattern will return to repitition. I'll have to keep throwing numbers in so that I don't end up with very (very) long repeating patterns. But in an infinite series of numbers, I would exhaust every possible combination of numbers in order to ensure it never becomes just a repeating sequence.

Does this not mean that, in an infinite sequence of numbers, every sequence of numbers of any length must occur?

 

[Office_Shredder]

Frankly, I didn't see what you were getting at, no.

Ok, a simpler example, going back to coins again. We flip an infinite number of coins. One might want to say that eventually you will get 1 billion heads in a row... in fact you can prove that the probability of this occurring is 1.

But the sequence of tails, tails, tails, tails, tails,... ad nauseum is a valid sequence of infinite flips of a coin, even though you might think it will never occur. The problem is that events that have a probability of 0 can in fact occur when your sample space is large enough.

Now for some reason you have a special sequence of coin flips. You really want to know what the pattern looks like, but for some reason you can't just look up what the billionth coin flip is. We would be tempted to apply probability to the sequence... is there a point in which there are a billion heads in a row? Probability tells us that with a probability of 1 this occurs, and can even go so far as to predict approximately where this will happen.

This isn't a proof that such a sequence of heads will occur, because again, events with a probability of 0 can occur. But it gives us good intuition as to what to expect, and if it turned out later that our sequence did not have a billion heads in a row at some point, we would be very surprised.

To loosely see how this information might be useful, let's consider translating back to numbers.

With probability 1, a number picked at random will be normal, which means that it has a string of a billion 0's in it at some point. So you decide to pick a number at random. 1/7. Well, that happens not to have a string of a billion 0's. We're surprised by this, and decide to investigate further. We discover this whole class of rational numbers which mostly do not have this property of having a billion 0's in a row. Rational numbers, of course, are far more important than their relative sparsity in the set of real numbers would suggest, so we're glad to understand there is a smaller subset of the reals which deserves more study.

Of course, this is an obnoxious way to "discover" the rational numbers, but if it turned that pi was not normal, the same process might occur. We would be surprised to find this out, and ask how the heck that occurred; investigate pi to see if we can generalize its properties to find a larger class of numbers that are not normal. Then maybe we find out that e and the square root of two also fall under this special class of numbers, and eventually theorems concerning the properties of numbers that are similar to pi in this way come out and advance our knowledge of how numbers work.

To go one step further, if pi was in fact created as a random string of digits, we could predict approximately where the first string of a billion 0's will occur with high probability (again, you could get a weird sequence of digits). If we calculate pi out to that length and find that there is no string of anything close to a billion 0's, it might lead us to believe that pi is not actually normal. We can use this information to guide whether we attempt to prove that pi is normal, or that pi is not normal, potentially saving a lot of time

The distribution of prime numbers, and the prime number theorem that describes it roughly, have nothing to do with probability.

The probabilistic argument for why Goldbach's conjecture is probably true uses the prime number theorem to generate a probability distribution of prime numbers.

 

[DaveC426913]

Ok, a simpler example, going back to coins again. We flip an infinite number of coins. One might want to say that eventually you will get 1 billion heads in a row... in fact you can prove that the probability of this occurring is 1.

But it seems the analogy is still completely missing the point.

Coin flips are not deterministic. pi's digits are.

Every time I flip a coin, it will come up either heads or tails. That is true of the first flip, as well as the third, or the infiniteth flip.

Every time I test the first decimal of pi, it will always come up 1. That is true of the first decimal digit, as well as the 3rd .The inifiniteth digit will always be n (whatever n is).

Coin flips lead to red herring logic about pi.

 

[Office_Shredder]

but it seems you're still completely missing the point.

Coin flips are not deterministic. pi's digits are.

Every time I flip a coin, it will come up either heads or tails. That is true of the first flip, as well as the third, or the infiniteth flip.

Every time I test the first decimal of pi, it will always come up 1. That is true of the first decimal digit, as well as the 3rd .The inifiniteth digit will always be n (whatever n is).

Given the sequence of coin flips T,H,T,H,T,H,T,H,... that alternates between tails and heads every time, each time I ask what the first coin flip was I see that it's a tails.

Don't think of pi as being generated randomly, think of pi as being picked randomly from the set of all numbers. Of course pi is fixed, but the set of all numbers has a structure that means a number being picked at random from it almost certainly has certain properties. Similarly, I picked a sequence of heads and tails randomly from the set of all possible sequences, and for the purposes of making arguments, we consider that sequence as having been generated by flipping a coin a bunch of times

As another example, every time I ask what the first prime number is, it's two. No changing that, but that doesn't stop people from making long term probabilistic projections to consider whether things like Goldbach's Conjecture make sense.

 

[disregardthat]

3.1415926 ...2323234232323
Does this not mean that, in an infinite sequence of numbers, every sequence of numbers of any length must occur?

Of course not, consider the sequence 0,0,0,0...

Also, not every pattern in a decimal expansion results in a rational number. Consider the number

0.101001000100001000001..., where the number of consecutive zeros are increasing by one between each digit 1. This is an irrational with no repeating pattern, but obviously does not e.g. the sequence 222 occur in the expansion.

 

[DrRocket]

3.1415926 ...2323234232323

Better?

Hm. No. Now I'm having doubts again.

No matter how many times I throw a number 4 into the mix of 23s, eventually the pattern will return to repitition. I'll have to keep throwing numbers in so that I don't end up with very (very) long repeating patterns. But in an infinite series of numbers, I would exhaust every possible combination of numbers in order to ensure it never becomes just a repeating sequence.

Edit: Jay apparently punched "go" just ahead of me.

Does this not mean that, in an infinite sequence of numbers, every sequence of numbers of any length must occur?

No it does not.

There are infinitely long non-repeating patterns, and you only need two symbols to create one. Thusly:

0.01001100011100001111...

Note that there is a pattern in this case, but it is not a repeating pattern. There need not be any specific pattern at all, though this example does not demonstrate that.

Edit: Jarl apparently punched "go" just ahead of me.