The problem with probabilities is always, imo, semantic. We call the unknown "probability", for lack of a better way to describe things. But nothing is random, in reality. That's where the issue lies.

If I roll a die, I don't really have 1/6 chance of rolling a 6. I have either 0% chance, or 100% chance - depends on how I roll it. If I roll it the same way every time, I'll get the same result every time. Rolling a dice isn't "random". We call it random, because we don't know HOW we rolled it.

As for why this thing we call probability doesn't change, that's the easy part. A die has 6 faces, and if we could toss it randomly (we can't, randomness doesn't exist), any of them can come up, and they'll come up the same number of times (as in, given a truly random toss, any of them can come up as much as any other). So we say if we roll a d6, each outcome happens 1/6 of the times. Doesn't matter if you rolled it once, or 1000 times - each roll can come up as any number 1/6 of the time.