"If you just randomly read the output noise from an ADC it is possible, however unlikely, that you could read the integer sequence 1, 2, 3, 4, 5, 6, 7, 8, 9, 10."

Could you give us you mathematical definition for the randomness of that particular set of numbers?
This is the same as "enough monkeys with typewriters would eventually come up with all existing books"

Erik