Probability concept that distinguishes likelihood of sequences 0110101011101... and 000000000000...? Say we have a coin and want to decide if it is fair or not. We flip it many times. Consider two cases.


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*Say the result is a sequence like 0110101011101...

*The result is 000000000000...


In the first case the assumption that the coin is fair sounds reasonable, while in the the second the coin is obviously completely biased. However, for a truly fair coin the probabilities of the two sequences are the identical. What is then the concept that distinguishes the first one as more likely? How does one quantify that?

Consider also the third case


*The result is 010101010101010...


This actually does not look like a coin at all, but it seems to pass naive tests which boil down to comparing number of zeros to number of ones. Is there a sense in which it is less random then (1)? Or I have to invent a new rule if somebody gives me a more cleverly crafted sequence? 
 A: The sequences themselves do have the same probability. However, what makes the first seem more likely to you is the fact that it has a more even number of ones and zeros. Instead of testing the likelihood of the exact sequences, you can test the likelihood of the number of ones and the number of zeros. 
For a fair coin, this is binomial with $p = \frac{1}{2}$, meaning that the probability that the number of ones in a sequence of length $n$ is $k$ is 
$P_{\frac{1}{2}}(k) = {n\choose{k}}\frac{1}{2}^n$
The only thing that differentiates the sequences is essentially ${n\choose{k}}$, which represents the number of sequences of length n with k ones. This would indeed give a higher probability to your first sequence. 
Another way of looking at it, is treating $p$ as unknown. For a given $p$, such that $P(x_i = 1) = p = 1 - P(x_i = 0) \forall i$, the probability of a sequence $x_1, ... x_n$ is 
$P(x_1, ... x_n | p) = p^{ |\{i : x_i = 1\}|}\cdot (1 - p)^{|\{i : x_i = 0\}|}$ 
Now, which $p$ maximizes the likelihood for each sequence? How far is it from the true $p = \frac{1}{2}$ for a fair coin? 
A: You can distinguish your three sequences in terms of their period. 
The constant sequence $00000000...$, as well as $1111111111...$ have period $1$ and can be generated by the difference equation $a(n)=a(n-1)$. Closed forms for the sequence are $a(n)=0$ and $a(n)=1$
Sequences $010101...$ and $1010101...$ have period 2 and follow the recurrence $a(n)=a(n-2)$. The corresponding closed forms are $a(n)=\frac{1-(-1)^n}{2}$ and $a(n)=\frac{1+(-1)^n}{2}$. These are simple deterministic formulas, so these sequences are seen to be less random than the first (not random at all).
The patterns can be predicted linearly with zero prediction error after the initial values. One single value is enough for predicting sequence (2) and two values are needed for sequence (3), corresponding to the order of the linear models (first and second order).
The first sequence suggests no period. If it was periodic, the period would be at least $11$, eleven (the last two bits $01$ echoing the first two bits).

From another point of view, let us map your sequences to numbers. Interpret each bit in a positional system, each bit having weight $2^{-n}$, where $n$ is its position.
Your second sequence maps to the integer number $0$.
Your third sequence maps to the rational number $$\sum_{k=0}^\infty \frac{1}{2^{2k+1}}=\frac{2}{3}$$
The first sequence would likely map to an irrational number, unless some periodicity appeared later on.
Since most numbers are irrational (Probability of Getting a Rational Number), sequences like the first one are more likely.
A: As mentioned in the comment by Spencer, essentially the intuition comes from entropy and what you might call the micro-state/macro-state distinction. You actually have the answer in your question. I've highlighted it

  
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*Say the result is a sequence like 0110101011101...
  
*The result is 000000000000...
  

A sequence like 0110101011101 is way, way more likely than 000000000000. There's loads of sequences like that. 1011011101001 is one of them, 0101101010110 is another, and so on. My guess is that for well over 90% of sequences that long we'd both easily classify it as like your first sequence.
Now it could be that you chose your first sequence very carefully. It could be the precise launch sequence of a missile, for all I know, and you're sitting in a bunker bored out your mind at the moment. Now, if that is the case, you would say it's very, very unlikely to happen by a coin toss, right? What are the chances of that.
Similarly, for all you know, I'm the one sitting in the bunker and I've just spotted that you've written my launch code and I'm on the phone to the secret police to say someone's written our launch code on the internet. I mean, they say it's about tossing a coin but what are the chances of them choosing that randomly?
(Or it could be the digits of Pi, or "You Suck" in ASCII, or the name of an obscure Merzbow album, or ...).
So, the first thing you need to do when you're thinking about things is to group all of the precise sequences (the micro-state) into broad categories that you care about, know about, can know about, etc (the macro-states). Then you can assign a probability to each macro state.
The other answer suggest counting ones and zeros, that's a good start. But sequences like 0101010101010101010101 would be pretty freaky, too.
Underneath all this the space of detectable patterns to humans is much smaller than the configuration space of coin tosses and that in almost all cases the vast majority of configurations map to a pattern we call "random". So the probability of each pattern is very different. Exactly which depends a lot on the person but, you're right 0000000 probably goes to a pattern other than "random" with a small number of other configurations (and so a low probability) and 0110101011101 firmly into random (and so very likely).
As an aside, another way of looking at the pattern issue is to think of the sequences in terms of an approximation to their "Kolmogorov complexity". Think of how long computer code or a description of how to generate the sequence would be (in characters). The code or description for 00000000.. is going to be pretty short. Some (most!) are going to be so "random" that the only real way to describe them is just to write it down and say "print that". In that way you can put a precise figure on how "patterny" a sequence is. Because some sequences have incredibly short descriptions, by the pigeon-hole principle, most are going to end up in this "I give up" set. This isn't actually much more formally powerful than the former above, though, because it depends heavily on the definition of your instruciton set and the oracle of data it has available: it would easily capture any "mathsy" patterns we might spot (including most simple patterns) but probably not that your sequence is the opening lines to the declaration of independence in morse code or last week's lottery numbers. But it's a more precise definition for compuatation-oriented patterns.
