probability of a run of 100 6s in an infinite number of rolls of a die i came upon this problem that i couldn't solve they way i wanted. 
Basically, fair die is being rolled infinitely many times. Prove that the probability of there somewhere being 100 consecutive 6's is 1. 
I thought the answer would be like pi, i mean that there are all possible versions of numbers from 1 to 6 (in pi there would be any number combination where numbers are from 1- 9 because it runs infinitely). Could i also say that there is 100 consecutive 6's in pi somewhere, and the probability of it is 1 ? 
I'd like to think that i know the answer to this one, but i can't really prove it. 
As it is quite a famous problem , you can also tell me a book where i can find more information how to solve this ? 
 A: For rolling dice, you can prove the probability is $1$.  One way is to note that the probability that the first $100$ rolls are not all sixes is $1-\left(\frac 56\right)^{100}$, which is very slightly less than $1$.  The chance that rolls $101$ to $200$ are not all sixes is the same, $1-\left(\frac 56\right)^{100}$.  The same goes for each block of $100$  If we now consider $n$ blocks, so the rolls up to $100n$ the chance that we don't have a run from $100k+1$ through $100k+100$ that are all sixes is $\left(1-\left(\frac 56\right)^{100}\right)^n$, which we can make as small as we want by taking $n$ large enough.  I used disjoint blocks just to avoid worrying about overlap, but in fact the chance of having a run of $100$ sixes somewhere rises much faster than this.  For infinite rolls, then, the probability of not having a run of $100$ sixes is zero.  It is still possible that you don't.  
For $\pi$, we believe $\pi$ is normal, which implies that there is a run of $100$ sixes somewhere, in fact that they happen at exactly the expected density, but we haven't proven it.
