[prolyx]

Precisely. A difference only arises when comparing sets of strings. In our example, the number of elements in the set uniform strings aaaa...a of length L is equal to the number of distinct symbols N, while the number of district strings of length L is N^L. So if you ask "will the string be uniform?" (which is the only reason you thought up aaaa..a in the first place) you find the probability is exponentially small for long strings: N^(1-L). Really if you tend to choose a string for any particular reason other than chance, it can be exploited as long as the reason can be guessed (e.g. the dice rolls 31415 write out the digits of pi.).