Yesterday night I was watching a TV series, "Person of Interest" and there was this small wonderful speech about "pi" the irrational number in it.

This made me do a Google search to find some place where i can look-up my phone number in the decimals of pi and i reached this website http://www.angio.net/pi/piquery which allows us to do such a search in the first 200 million digits of pi. As such I was disappointed that I couldn't find my phone number of my wife's number. Anyway, after playing around with this application, i realized that numbers of length upto 8 digits can be found more or less in this app but finding a 9 digit number is rare and the we could just forget the 10 digit numbers. This made me think of a question (i am sure mathematicians must have wondered about this question many times before) that is it possible to build/determine a probability distribution function which can tell us that probability of finding X (some number with m digits) in the first N digits of pi is p(X).

I definitely have no idea about how to approach this problem. But would love to know whether such a function can exist, and if it exists then how or can it be determined (even in some limited form, let's say for the first billion digits of pi, etc), and if answer to that too is yes then what is that probability function? Would love to see any leads for this interesting nerdy problem :)