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We know that Pi is a pseudo random sequence that continues indefinitely, so we know that there is a million consecutive ones(or any other combination) contained within Pi somewhere. So then, if we were to start at any random number, say the 978th, how many digits of Pi would you have to search through to get a statistical 50% chance of getting a million ones consecutively? How would increasing it to 50 million ones consecutively change it? How would increasing the chance to 60% or 90% change it?

I've been thinking about for a couple of days, but no clear way to solve it has come to mind. I was thinking about enumerations and powers of ten, but I can't think of a way to incorporate the chance in.

Also, in the first two hundred million digits of Pi, there isn't even nine consecutive ones once! (www.angio.net/pi/). And, is it even containable within 2^64?

EDIT: This question, now that I think about it, would be far more appropriate assuming that each digit was decided by rolling a ten faced dice. I don't want to get into the depths of Pi itself as that is not necessarily my question. Thanks!

At the start of your post you say we know a lot of things; in fact we don't know those things.

Actually various of your assertions are meaningless, at least until you supply a few relevant definitions. But never mind that - the question is obliterated by one fact: No, we do not know that there are a million consecutive ones somewhere in the digits of $pi$. Hence we can't say much about how long you should expect to have to look.

The edit substantially rehabilitates your post: The question of how far we'd need to search a random sequence of digits before finding a million consecutive ones makes perfect sense. Did states that the answer is of the order of $omega=10^{1,000,000}$ - don't hold your breath.

I decided to post an answer when I realized that there is something one can say about all this other than just no, no, no. At the end you express surprise that the first $10^8$ digits do not contain nine consecutive ones. In fact this is not surprising at all. There are $10^9$ nine-digit sequences, and fewer than $10^8$ nine-digit sequences appear in the first $10^8$ digits of $pi$. So if you choose a nine-digit sequence at random the probability that it appears in the first $10^8$ digits of $pi$ is less than $1/10$.

Analyzing things that way it's clear that if it is true that every million-digit sequence appears in the expansion of $pi$ we need to look at at least $omega$ digits to verify this fact by brute force; it seems iikely that the actual number is much larger, "surely" some sequences will not appear until much later than expected.

In particular it's possible that we will never know whether a sequence of a million ones appears. Because we will never build a computer with an $omega$-byte memory; the number of electrons in the observable universe is infinitesimal compared to $10^{1,000,000}$. Hmm. Ok, we could search for that one specific sequence using less than $omega$ bytes of memory. But if we do a billion billion billion petaflops it still takes essentially $omega$ seconds; the universe has got to its "heat death" or "big crunch" long before then.

(That's assuming a "classical" computer. Quantum-mechanical improvement by a factor of $10^{1000}$ makes essentially no difference; $10^{999,000}$ is still big.)