The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of William Shakespeare.

Source: http://en.wikipedia.org/wiki/Infinite_monkey_theorem

My question is actually if pi is a "random" sequence of numbers. (I've read in other posts that it's likely to be such a sequence, and that e.g. irrationality isn't a sufficient condition, as can be seen in 0.01001000100001...) But is there an elegant mathematical way to proof if a number is such a random sequence? Or can it be proven with statistics/numerical methods with a kind of certainty that it's such a number?

EDIT: people stated that the exact mathematical term for "random" that I was searching for, should've been "normal". So my questions boils down to:

Pi is likely to be a normal number; if it is, it contains every sequence of numbers. But if it isn't, does it then contain every sequence of numbers (although in this case not with the same likelihood)? Or is this not sure (or even a contradiction)?

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    It is conjectured that it is a http://en.wikipedia.org/wiki/Normal_number, but it remains an open problem – chubakueno Jul 10 '13 at 21:15
  • Please ask clearly the question you want to ask. The question about monkeys has an answer (since the strings produced by a countable number of monkeys are countable, the chance of producing a particular real number is zero). It is possible to define "random" in the kind of sense you mean (look for information about "normal numbers") - is that what you are looking for. It is not known whether $\pi$ is normal. Three questions. Very different kinds of answer. Ask the one you want and someone will help you with it. – Mark Bennet Jul 10 '13 at 21:15
  • @MarkBennet Yeah, I misinterpreted the question that way. He's talking about whether $\pi$ is a monkey, not if infinitely many monkeys would produce $\pi$. – Thomas Andrews Jul 10 '13 at 21:16
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    I think any number that we can prove to be normal (as of now) is specifically *constructed* to be normal (and the digit sequence looks much less random than you may expect). The probability that a, well, random real number is normal is $1$ from the beginning; statistical methods (especially if they work with only finitely many (and be it $10^{10^{10}}$) digits) cannot increase that suspicion. – Hagen von Eitzen Jul 10 '13 at 21:16
  • @MarkBennet "My question is actually if pi is a random sequence of numbers [...] elegant mathematical way to proof". But how can you say that the chance of producing a particular real number is zero, if every combination of digits is is equally likely to occur? E.g. the chance that you'll encounter the "real number" 9 in the monkey's text (or in pi) is 100%? – BNJMNDDNN Jul 10 '13 at 21:26
  • I think some of the trouble here is what you mean by "random". $\pi$ is a fixed constant; all digits are pre-determined by any one of a number of ways to calculate $\pi$. So you need to clarify what you mean by "a random sequence of numbers" and how you hope $\pi$ will live up to that. – Arthur Jul 10 '13 at 22:52
  • @BNJMNDDNN If that is your question, put it in the title. And several of us have pointed you to "Normal numbers" - if that is not what you want you will have to explain further. – Mark Bennet Jul 11 '13 at 05:22
  • I edited my post a little. I've come already a lot wiser out of your comments, but I'm still left with the question: if pi is not a normal number, does it then contain all possible sequences of numbers? In other words: can it be stated with 100% certainty that pi contains all possible sequences of numbers? – BNJMNDDNN Jan 20 '14 at 15:07

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