Why is the probability of finding a winning nonce exactly the same for a new header and a header where you've already tried 1 billion nonces? I'm referring to this question: When do miners stop waiting for new transactions?

More specifically, this statement from the top answer: 

"suppose you have a block header h1 on which you have already tried a billion nonce values, and a header h2 which has just been generated and on which no nonces have been tried yet. If you have the choice as to which one to hash next, which should you choose, for the greatest chance of finding a winning nonce? The answer is, it makes no difference - they both have exactly the same probability. So there is no harm in switching to h2"

I would have thought that the probability would be slightly higher for the first block (h1) as there are 1 billion nonces that you know do not succeed, i.e. fewer winning possibilities.

Complete newbie here and to the bitcoin world so apologies if I'm missing something obvious.
I would have thought that the probability would be slightly higher for the first block (h1) as there are 1 billion nonces that you know do not succeed, i.e. fewer winning possibilities.


For every computation, you have the same probability of generating a valid block hash. Nothing you can do will change this, even if you try every possible nonce value for a given header. Nothing about what you find while testing nonces will tell you anything about what the next nonce value will return as output. Each guess does not get you ‘closer to the goal’, it is just a random chance.

Think of it like this: 


You have a six-sided die (numbered 1 to 6) that you know is fairly weighted. If you roll a 1, you get to make the next block! Every other roll does nothing. 
You roll the die ten times, and don’t get a 1 on any of the rolls. 
Now, on your eleventh roll, you are given the option to switch dice. You can replace the one you were rolling, with a new one that is otherwise the same (also six-sided, printed with 1−6, fairly weighted). Your choice.


What will give you the better odds of rolling a 1? The answer is: it doesn’t matter. The fact that the first die didn’t roll a 1 yet doesn’t influence what the next roll will be. The same goes for the replacement die. Just because it hasn’t been rolled yet, doesn’t make it any more likely to be a 1 (or any other number).