Congrats on discovering and demonstrating that there's a 63% chance of success to hit X at least once, when p=1/N, in N tries, btw. No one really bothered to do the math on that one. You're definitely the first.
So then, if you're saying the prefix method is mathematically sound, right? Why are you giving it shit?
N balls in an urn. one is red, the rest are blue.
Extract one ball, if it's blue = success. If it's red: failure.
Put the ball back in the urn.
This gives you an exact 1:1 analogy with either searching for an H160 full hash, or whatever prefix size of it you desire. You can use any other classification criteria as you see fit, the analogy doesn't change.
p = 1/n
trials = n
Probability of failure per extraction: (n-1)/n = 1 - 1/n
Total probability of failure over N extractions: (1 - 1/n) ** n
Probability of success to get the blue ball at least once over N extractions:
p_success = 1 - probability_failure = 1 - (1 - 1/n)**n
This function converges towards 1 - 1/e = 63.212...% as N goes to infinity. It's a very exact number to calculate.
Now: what in the freaking world changes after you put the extracted ball back in the urn?
Answer: NOTHING. That's why the prefix theory is total bullshit, and is one of the reasons this thread is junk science, which I'm retreating from. Feel free to go and pat mcdouglasx, he needs a hug.