Probability of finding a small number within a much larger one

[Hans Havermann]

I wrote:

> Specifically, I've determined that 10000013 is the smallest length-8 
> number not found embedded in the 369693100 digits of 9^(9^9). Could 
> someone with a good grasp of probability make the case that the 
> time-consuming search of length-7 numbers is not likely to yield a 
> smaller result?

Well, maybe it's not as difficult as I had imagined. Assuming random 
distribution of digits... ninety million length-8 numbers yield an 
expectation of only 4 hits for each one, whereas nine million length-7 
numbers yield an expectation of 41 hits for each one. The former is 
close enough to zero to fail for a fraction of those numbers, whereas 
the latter would require a sizable deviation.