(I will start this by saying that I kinda disagree with every commenters opinion in this chat.)
I also want to say that this is an opinion and I’m not a statiscian. You’ll find that in a lot of higher level math, everything goes from being universally accepted back to being 1st grader pissy fights over who’s right and who’s wrong.
Your on the right track, which is that the argument that prior rolls have no effect on future rolls doesn’t hold water when you actually think about it. It could be that theirs some nth dimensional reasoning that makes everything makes sense, but to the best of my knowledge no one’s figured it out yet.
Let’s imagine a test, where every roll of the die is completely independent of every other roll. Your running this test, and you decide to do 100 rolls of the die.
You roll a 1 on the first roll. What luck! Is your probability of rolling a 1 on the second roll still 1/20? No, it is in fact 1/400 (because probabilities multiply - point is the chances are much slimmer), and I’ll explain why.
Well I don’t really know why but I can explain why the other logic is stupid! Basically if we say that there’s no change to probability, we would end up with some really weird and easily disprovable conclusions. For starters, we would never expect a distribution across our dataset. It would be a 1/20 chance that all 100 of rolls rolled a 1, instead of a 1/1.2E130 that it actually is. It would be superrrr weird and if you rolled the die 100 times and only ever rolled a 1 right? But if were saying that our probability doesn’t change, then you should only need to repeat the test around 20 or so times to get the desired 100 1’s. Even as a thought experiment, everything starts to break down.
I think (and I may be wrong about this) it has to do with the nature of branching possibilities. Essentially the way these are figured out is you take the all of the possible outcomes and you say what’s the chance of this one particular outcome. Which is just 1/all of the possibilities (in a 20 sided die this is 20). When you start having multiple independent events, you get much more then just 40 or 60 possibilities. Every roll branches into the next, so the outcome of our first roll branches into the next 20 possibilities our second roll. When you consider all the possibilities of the first and second roll you end up every with every possibile role having another 20 possibile outcomes. Basically this works out to 20*20=400 possibile outcomes (this is where the 1/400 came from earlier). This continues into the the third roll with 203 = 8000 possibilities, of which a triple 1 is only 1 possible branch ergo it’s probability is 1/8000.