By the way, I just ran a simulation (using C++ since I suck at Java) and it looks like my random number distributor is pretty uniform.
TOTAL:
# of 0's: 226
# of 1's: 232
# of 2's: 208
# of 3's: 247
# of 4's: 230
# of 5's: 227
# of 6's: 240
# of 7's: 208
# of 8's: 226
# of 9's: 216
EDIT:
Dang it! There's a little repetition. I noticed that a 0 is appearing every 50th character eventually in the output. My formula for the nth number is f(n) = (n * ∏(n)) mod 10.
5, 8, 5, 8, 9, 2, 7, 0, 1, 4, 3, 2, 5, 8, 3, 8, 3, 0, 3, 8, 9, 6, 5, 6, 1, 6, 1, 0, 7, 2, 1, 6, 5, 6, 9, 4, 3, 0, 9, 2, 3, 2, 5, 4, 7, 4, 3, 0, 3, 8, 3, 4, 5, 8, 3, 8, 3, 0, 3, 6, 1, 4, 5, 2, 7, 6, 3, 0, 3, 8, 1, 2, 5, 8, 3, 6, 9, 0, 9, 2, 3, 2, 5, 8, 3, 6, 9, 0, 7, 8, 1, 4, 5, 8, 3, 8, 7, 0, 7, 4, 9, 6, 5, 2, 9, 4, 1, 0, 7, 6, 1, 6, 5, 6, 1, 6, 7, 0, 1, 6, 1, 2, 5, 6, 3, 2, 3, 0, 9, 6, 3, 8, 5, 4, 1, 6, 3, 0, 1, 2, 9, 8, 5, 4, 1, 6, 1, 0, 7, 2, 9, 8, 5, 6, 3, 2, 3, 0, 7, 6, 1, 4, 5, 8, 7, 6, 1, 0, 9, 2, 3, 2, 5, 4, 7, 8, 7, 0, 7, 2, 9, 8, 5, 4, 9, 4, 1, 0, 3, 6, 3, 4, 5, 8, 7, 4, 3, 0, 9, 2, 1, 6, 5, 2, 3, 4, 1, 0, 7, 2, 9, 8, 5, 6, 9, 8, 3, 0, 1, 8, 7, 2, 5, 4, 1, 2, 3, 0, 3, 8, 3, 4, 5, 2, 3, 4, 1, 0, 3, 2, 9, 6, 5, 4, 9, 8, 1, 0, 7, 2, 1, 6, 5, 4, 7, 6, 3, 0, 3, 4, 7, 2, 5, 4, 3, 8, 7, 0, 1, 4, 9, 6, 5, 8, 9, 2, 9, 0, 3, 2, 1, 4, 5, 6, 1, 6, 1, 0, 1, 4, 9, 4, 5, 4, 7, 4, 1, 0, 3, 4, 7, 2, 5, 2, 9, 2, 1, 0, 3, 8, 3, 2, 5, 4, 3, 8, 7, 0, 1, 4, 3, 2, 5, 6, 3, 4, 3, 0, 1, 4, 7, 2, 5, 2, 3, 4, 9, 0, 3, 4, 7, 4, 5, 4, 7, 6, 9, 0, 1, 4, 3, 2, 5, 8, 3, 8, 3, 0, 7, 2, 1, 6, 5, 8, 9, 4, 9, 0, 9, 6, 7, 4, 5, 4, 7, 4, 1, 0, 1, 6, 1, 8, 5, 8, 7, 6, 7, 0, 9, 6, 7, 8, 5, 8, 3, 2, 9, 0, 9, 6, 1, 8, 5, 6, 9, 8, 7, 0, 3, 4, 9, 4, 5, 6, 3, 6, 3, 0, 9, 4, 1, 6, 5, 2, 1, 2, 9, 0, 1, 2, 7, 2, 5, 6, 3, 8, 3, 0, 3, 8, 7, 6, 5, 2, 1, 6, 1, 0, 7, 2, 9, 6, 5, 6, 3, 8, 7, 0, 9, 2, 7, 4, 5, 8, 3, 4, 1, 0, 3, 4, 7, 4, 5, 8, 3, 8, 3, 0, 3, 8, 1, 4, 5, 2, 3, 8, 1, 0, 7, 4, 7, 2, 5, 6, 9, 6, 1, 0, 1, 6, 9, 8, 5, 2, 1, 8, 3, 0, 9, 2, 9, 8, 5, 2, 7, 2, 1, 0, 3, 8, 3, 8, 5, 4, 1, 6, 7, 0, 3, 6, 1, 4, 5, 8, 9, 8, 1, 0, 1, 4, 7, 2, 5, 6, 1, 6, 3, 0, 1, 6, 1, 2, 5, 6, 9, 2, 9, 0, 3, 8, 3, 2, 5, 6, 9, 2, 7, 0, 3, 4, 7, 8, 5, 6, 9, 2, 1, 0, 1, 6, 9, 6, 5, 6, 1, 4, 1, 0, 7, 4, 7, 6, 5, 8, 1, 8, 3, 0, 1, 6, 3, 8, 5, 2, 1, 8, 7, 0, 7, 6, 1, 6, 5, 2, 3, 8, 3, 0, 1, 4, 9, 4, 5, 4, 1, 2, 9, 0, 3, 6, 7, 4, 5, 6, 1, 2, 7, 0, 9, 6, 1, 6, 5, 4, 7, 4, 3, 0, 1, 2, 1, 6, 5, 4, 3, 8, 7, 0, 3, 2, 1, 8, 5, 8, 9, 4, 1, 0, 9, 2, 3, 2, 5, 6, 3, 6, 9, 0, 9, 2, 1, 2, 5, 2, 1, 6, 9, 0, 9, 4, 7, 8, 5, 8, 9, 8, 7, 0, 1, 4, 9, 6, 5, 8, 9, 8, 9, 0, 3, 8, 1, 2, 5, 4, 7, 6, 9, 0, 9, 2, 1, 6, 5, 6, 1, 6, 1, 0, 7, 2, 1, 2, 5, 8, 7, 8, 7, 0, 1, 4, 3, 2, 5, 4, 3, 4, 1, 0, 1, 2, 1, 8, 5, 8, 3, 2, 7, 0, 1, 4, 9, 4, 5, 4, 1, 2, 9, 0, 1, 4, 7, 6, 5, 2, 1, 4, 3, 0, 3, 2, 1, 8, 5, 4, 9, 4, 3, 0, 7, 6, 7, 6, 5, 6, 1, 8, 9, 0, 3, 2, 9, 2, 5, 4, 1, 8, 3, 0, 9, 4, 1, 2, 5, 6, 9, 6, 1, 0, 1, 4, 9, 6, 5, 8, 1, 2, 9, 0, 3, 8, 7, 8, 5, 2, 3, 8, 1, 0, 1, 2, 3, 6, 5, 6, 1, 4, 1, 0, 7, 2, 7, 8, 5, 2, 1, 2, 9, 0, 9, 8, 3, 4, 5, 4, 3, 4, 3, 0, 3, 8, 3, 4, 5, 8, 1, 6, 1, 0, 1, 4, 9, 6, 5, 8, 3, 6, 9, 0, 7, 8, 7, 4, 5, 6, 9, 4, 9, 0, 1, 6, 7, 8, 5, 8, 9, 2, 1, 0, 9, 2, 1, 6, 5, 4, 9, 6, 7, 0, 1, 6, 7, 6, 5, 8, 9, 4, 7, 0, 7, 8, 3, 4, 5, 4, 7, 2, 7, 0, 7, 6, 3, 8, 5, 2, 7, 6, 1, 0, 7, 4, 9, 6, 5, 6, 9, 2, 1, 0, 3, 4, 9, 6, 5, 8, 9, 8, 1, 0, 9, 8, 9, 4, 5, 6, 3, 4, 3, 0, 7, 2, 9, 8, 5, 4, 1, 6, 7, 0, 1, 6, 1, 2, 5, 4, 1, 8, 3, 0, 7, 6, 1, 6, 5, 8, 1, 2, 9, 0, 9, 6, 7, 6, 5, 2, 3, 4, 9, 0, 7, 6, 1, 4, 5, 6, 3, 8, 1, 0, 1, 2, 9, 8, 5, 8, 3, 4, 3, 0, 3, 4, 3, 8, 5, 8, 1, 2, 3, 0, 9, 8, 9, 6, 5, 8, 9, 8, 3, 0, 3, 2, 1, 8, 5, 8, 1, 4, 9, 0, 9, 8, 9, 8, 5, 6, 9, 4, 1, 0, 1, 8, 9, 6, 5, 8, 3, 8, 3, 0, 7, 6, 3, 6, 5, 8, 9, 2, 9, 0, 7, 8, 7, 4, 5, 2, 9, 4, 7, 0, 3, 2, 3, 2, 5, 6, 3, 8, 3, 0, 3, 8, 3, 2, 5, 4, 9, 8, 7, 0, 7, 6, 1, 4, 5, 6, 1, 8, 3, 0, 3, 8, 3, 8, 5, 6, 9, 6, 9, 0, 3, 2, 7, 2, 5, 6, 1, 2, 9, 0, 1, 4, 9, 2, 5, 6, 7, 4, 3, 0, 1, 6, 1, 2, 5, 6, 9, 8, 9, 0, 9, 4, 3, 8, 5, 8, 3, 4, 1, 0, 3, 8, 3, 8, 5, 4, 9, 2, 9, 0, 9, 6, 7, 8, 5, 6, 9, 8, 7, 0, 7, 8, 9, 6, 5, 2, 3, 8, 7, 0, 1, 4, 9, 6, 5, 6, 3, 6, 7, 0, 9, 4, 7, 4, 5, 4, 3, 8, 7, 0, 3, 2, 9, 6, 5, 6, 3, 4, 1, 0, 1, 6, 3, 2, 5, 2, 1, 2, 3, 0, 3, 8, 3, 2, 5, 6, 7, 4, 3, 0, 7, 8, 3, 6, 5, 4, 1, 4, 3, 0, 3, 4, 1, 6, 5, 8, 3, 8, 1, 0, 9, 4, 3, 8, 5, 2, 7, 2, 9, 0, 3, 4, 1, 4, 5, 6, 1, 2, 7, 0, 3, 2, 3, 6, 5, 6, 9, 6, 7, 0, 1, 4, 9, 6, 5, 8, 3, 6, 1, 0, 7, 6, 7, 8, 5, 8, 9, 6, 3, 0, 9, 2, 9, 8, 5, 8, 9, 2, 9, 0, 9, 2, 3, 2, 5, 2, 1, 2, 7, 0, 7, 2, 9, 6, 5, 4, 7, 6, 9, 0, 1, 6, 7, 2, 5, 4, 7, 4, 3, 0, 7, 4, 3, 2, 5, 6, 9, 4, 1, 0, 9, 2, 7, 8, 5, 2, 9, 8, 7, 0, 7, 2, 7, 6, 5, 2, 3, 8, 7, 0, 9, 6, 7, 4, 5, 8, 7, 6, 9, 0, 9, 6, 1, 8, 5, 8, 3, 4, 3, 0, 9, 2, 9, 6, 5, 6, 7, 6, 3, 0, 7, 2, 9, 6, 5, 8, 9, 8, 3, 0, 3, 8, 9, 6, 5, 6, 1, 6, 7, 0, 7, 8, 3, 4, 5, 8, 3, 6, 9, 0, 1, 6, 7, 2, 5, 8, 9, 2, 9, 0, 9, 6, 3, 2, 5, 6, 9, 4, 1, 0, 1, 4, 9, 4, 5, 2, 3, 8, 3, 0, 9, 4, 9, 6, 5, 6, 1, 2, 3, 0, 7, 6, 7, 4, 5, 2, 1, 4, 9, 0, 3, 4, 9, 4, 5, 4, 3, 8, 7, 0, 1, 6, 7, 2, 5, 2, 7, 6, 1, 0, 1, 6, 7, 8, 5, 4, 1, 8, 7, 0, 9, 6, 1, 2, 5, 8, 1, 2, 7, 0, 1, 4, 1, 4, 5, 8, 7, 6, 7, 0, 7, 8, 7, 4, 5, 4, 9, 8, 3, 0, 3, 4, 3, 8, 5, 2, 7, 6, 1, 0, 7, 2, 3, 4, 5, 8, 3, 6, 3, 0, 3, 6, 1, 2, 5, 2, 7, 6, 7, 0, 7, 6, 7, 6, 5, 6, 9, 8, 7, 0, 9, 2, 3, 2, 5, 6, 9, 2, 7, 0, 9, 8, 1, 6, 5, 6, 3, 4, 3, 0, 3, 2, 9, 8, 5, 6, 7, 4, 7, 0, 7, 8, 7, 4, 5, 8, 7, 6, 9, 0, 3, 4, 1, 2, 5, 8, 3, 4, 3, 0, 7, 2, 9, 6, 5, 6, 1, 4, 9, 0, 7, 2, 1, 6, 5, 4, 7, 6, 9, 0, 3, 2, 1, 8, 5, 6, 3, 2, 7, 0, 3, 4, 9, 6, 5, 2, 7, 4, 9, 0, 1, 4, 7, 2, 5, 8, 9, 2, 7, 0, 3, 4, 9, 4, 5, 6, 9, 4, 1, 0, 1, 8, 9, 8, 5, 6, 9, 8, 7, 0, 7, 8, 7, 8, 5, 2, 1, 2, 1, 0, 1, 4, 7, 2, 5, 2, 7, 4, 9, 0, 7, 6, 3, 4, 5, 6, 7, 6, 1, 0, 1, 6, 1, 2, 5, 2, 9, 4, 9, 0, 3, 8, 3, 8, 5, 4, 3, 6, 7, 0, 9, 2, 7, 6, 5, 8, 9, 4, 9, 0, 3, 4, 7, 2, 5, 6, 9, 4, 7, 0, 1, 4, 3, 4, 5, 4, 7, 4, 9, 0, 9, 8, 3, 2, 5, 4, 7, 6, 9, 0, 3, 4, 3, 2, 5, 4, 3, 8, 3, 0, 7, 2, 1, 2, 5, 2, 1, 2, 3, 0, 7, 8, 3, 8, 5, 4, 1, 6, 1, 0, 3, 2, 3, 8, 5, 2, 7, 4, 3, 0, 3, 4, 3, 8, 5, 6, 9, 2, 9, 0, 7, 8, 9, 8, 5, 4, 7, 2, 7, 0, 3, 2, 7, 2, 5, 4, 7, 4, 1, 0, 1, 4, 3, 8, 5, 2, 7, 4, 3, 0, 7, 6, 7, 6, 5, 8, 7, 6, 7, 0, 7, 8, 3, 2, 5, 6, 9, 8, 7, 0, 3, 8, 9, 6, 5, 2, 3, 4, 3, 0, 3, 8, 3, 2, 5, 4, 1, 2, 9, 0, 1, 4, 3, 8, 5, 8, 3, 8, 3, 0, 1, 2, 1, 6, 5, 6, 9, 4, 3, 0, 9, 4, 1, 2, 5, 4, 9, 4, 1, 0, 3, 4, 7, 4, 5, 2, 7, 4, 1, 0, 1, 4, 9, 4, 5, 4, 9, 4, 3, 0, 3, 2, 1, 6, 5, 8, 3, 6, 9, 0, 3, 8, 3, 8, 5, 4, 3, 8, 1, 0, 9, 2, 1, 6, 5, 8, 9, 6, 1, 0, 1, 6, 1, 2, 5, 2, 9, 2, 9, 0, 9, 2, 1, 8, 5, 4, 7, 4, 9, 0, 3, 8, 3, 4, 5, 4, 3, 6, 1, 0, 7, 2, 1, 6, 5, 6, 1, 6, 1, 0, 1, 4, 7, 8, 5, 6, 9, 8, 7, 0, 9, 8, 3, 4, 5, 6, 9, 4, 1, 0, 1, 6, 1, 6, 5, 4, 7, 6, 9, 0, 9, 2, 9, 8, 5, 4, 9, 8, 7, 0, 3, 8, 9, 4, 5, 6, 7, 4, 7, 0, 1, 6, 7, 4, 5, 8, 9, 4, 3, 0, 1, 2, 7, 4, 5, 8, 3, 8, 7, 0, 7, 6, 1, 2, 5, 8, 3, 2, 1, 0, 1, 6, 1, 6, 5, 4, 1, 2, 9, 0, 3, 2, 1, 4, 5, 8, 3, 8, 3, 0, 9, 2, 1, 2, 5, 8, 9, 8, 7, 0, 1, 4, 7, 6, 5, 8, 3, 8, 1, 0, 3, 6, 7, 4, 5, 6, 1, 6, 9, 0, 7, 8, 9, 4, 5, 2, 1, 2, 9, 0, 3, 8, 3, 8, 5, 6, 1, 4, 1, 0, 7, 6, 1, 2, 5, 6, 7, 6, 3, 0, 9, 4, 1, 6, 5, 8, 3, 4, 3, 0, 1, 6, 7, 4, 5, 8, 7, 6, 1, 0, 3, 8, 1, 8, 5, 6, 1, 4, 1, 0, 3, 4,