[Someone]

Amusing fact: the probability that a sequence can be found is not equal for all sequences.

A simple example: 2 binary digits in binary sequences of length 3. There are eight binary sequences of length 3:

  000
  001
  010
  011
  100
  101
  110
  111

3 of those contain '00' but 4 of them '01'. Reason for the discrepancy is that one of those with '00' has 2 overlapping occurrences, but is counted only once. You get this as soon as overlap can occur, i.e. when the sequence to be found starts with x digits that it also ends with.

Of course, none of this matters, especially not when d << N, which it will be if N goes to infinity.

Also, the mathematical term is 'normal number' (http://mathworld.wolfram.com/NormalNumber.html), and we do not know whether pi is normal.