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Is there any way to estimate how "deep" we'd need to go into the decimal expansion of a normal number to find a specific string of digits, say a 10-digit pattern?

What about a 100-digit pattern?

BradC
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    Sure: an $n$-digit pattern has probability $10^{-n}$, so the expected number of digits you'd have to go through to see one is about $10^n$. – Qiaochu Yuan Oct 15 '15 at 18:01
  • @QiaochuYuan Ok, so you'd expect to slog through a literal google (10^100) arbitrary digits to find the specific 100-digit string you're looking for? That makes the monkey/typewriter problem more of a needle/haystack problem... – BradC Oct 15 '15 at 18:05
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    @BradC The point of the monkey typewriter problem is not that Shakespeare is easy, it is that infinity is really, really big. :) – Thomas Andrews Oct 15 '15 at 18:07
  • @ThomasAndrews Right. I'm answering a question on another site about whether pi contains "infinite information" and if that is impossible or weird in some way. Setting aside the fact that it is unknown whether pi is or is not normal, I'm looking for a way to show how pretty much useless the "information" really is. I think the impracticality of searching through 10^100 digits helps, thanks. – BradC Oct 15 '15 at 18:10
  • How could it be any less - there are $10^{100}$ sequences of one hundred digits, so some hundred digit sequence can't occur earlier than $10^{100}$ places out, so the expected number if the normal number is random in at least $5\cdot 10^{100}$. Essentially, to add a digit, it takes you ten times longer to find a match. (In the same way, if you were flipping a coin repeatedly, the number of times you'd expect before you got a specific ten-toss pattern would be $2^{10}$.) – Thomas Andrews Oct 15 '15 at 18:19
  • Also, see this old question/answer. http://math.stackexchange.com/questions/431567/does-pi-contain-the-combination-1234567890/431601#431601 – Thomas Andrews Oct 15 '15 at 18:22
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    Finally, normal does not mean random. It is possible for a normal number to be entirely predictable. $0.012345678910111213\dots$ is a normal real number that you can predict (with some care) how long it will take before you get any $n$ digits in a row. A little hard to write the formula, but for each instance you can make an easy argument. – Thomas Andrews Oct 15 '15 at 18:25
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    Depending on definitions, no real number we can describe in a finite amount of time contains infinite information. – Thomas Andrews Oct 15 '15 at 18:27

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