-snip-
And again, when you change your key or they change theirs, you are switching to a different system too.
The probability should be the chance of getting a specific outcome among other possible ones from THE SAME SYSTEM and PrimeDice obviously lack this.
If you want to write it like such then take a view at the bigger system outside of this.
Each player is given a randomly generated server seed and randomly generated client seed. Upon the change of a client seed, the server seed is again randomly generated, hashed and spit out to give to the user.
We can consider each of your systems a 'result' since it will return the same value. Thus with the randomness of generation, each user has the same probability of getting the same results.
Consider this, as well. There's a finite number of permutations for server and client seed. Each server and client seed combination results in a chain of results for wagering (because of nonces). That means that you're randomly given a chain of results. Pass this kind of thinking to as if you were rolling a die with a series of numbers on it.
When you roll a number, the probability of drawing the next number in the next roll should becomes lower.
On a roulette, the first time you spin, each number has 1/37 chance.
But the next spin, getting a number after that specific number becomes 1/(37)^2, because it is the same system.
But if you decided to go to a different table to spin a number, then it is 1/37 again.
So, 37 million spins on the same table you would expect around 1 million occurrence for each number
but if you did 37 million spins on 37 million different tables, there will be no "equal" distribution of occurrences for all numbers