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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)?


marked as duplicate by Joe Johnson 126, Joel Reyes Noche, dustin, user147263, Jonas Meyer Feb 17 '15 at 2:19

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    $\begingroup$ It is conjectured that it is a en.wikipedia.org/wiki/Normal_number, but it remains an open problem $\endgroup$ – chubakueno Jul 10 '13 at 21:15
  • $\begingroup$ 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. $\endgroup$ – Mark Bennet Jul 10 '13 at 21:15
  • $\begingroup$ @MarkBennet Yeah, I misinterpreted the question that way. He's talking about whether $\pi$ is a monkey, not if infinitely many monkeys would produce $\pi$. $\endgroup$ – Thomas Andrews Jul 10 '13 at 21:16
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    $\begingroup$ 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. $\endgroup$ – Hagen von Eitzen Jul 10 '13 at 21:16
  • $\begingroup$ @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%? $\endgroup$ – BNJMNDDNN Jul 10 '13 at 21:26