alt Hacker News> The probability of finding a block is always the same, given a hashrate.
I think you are missing something very basic here: the longer you compute, the higher the likelihood that you will find the hash before the others.
The extreme case being that if you can try ALL the possibilities before the others can start, then you are guarantee to find the solution before them.
That's only mathematically true. The advantage is way too small to be relevant.
Your advantage is having exhausted a fraction of the search space. But that fraction is tiny.
You're trying to find a hash with a value below a certain threshold (simplified said, a hash starting with a certain amount of zeroes). You do this by trying random inputs to the hash function. Every input has the same probability of getting an output that is low enough in value. You are not advancing by having tried other inputs. It's practically equivalent to rolling multiple dices until enough of them show a one. Every roll has the same probability of succeeding regardless of the rolls before.