I think you may be mixing probability with frequency here.

If you say "I don't know the answer, but I will use heuristic reasoning to make a guess" and you ask for the ones digits of the trillionth, trillion and third, and two-trillion-and-seventeenth primes, you expect to get three numbers picked at random from that set, and guessing this way will serve you well.

You should rather say that they're independent if you're computationally limited. But that independence doesn't even matter for this post. I think you're assigning too many things to the word "random." Probability is not about random, it's about ignorance.

EDIT: speaking of ignorance, I probably made this comment in ignorance of what you actually intended, see more recent comment.