then it's 4% each result. you don't want to know what happens with the missing 20%.

The standard answer is that the odds of the first roll don't change the odds of the second roll, the second roll still has a 1/20 chance of a 1, no matter what the first roll is. The more thorough answer is that it's a misunderstanding of what probabilities are. Yes, there's a 1/400 chance of rolling 2 1s, but by the time you roll the first die and get a 1, you're not talking about that problem anymore. You've introduced new information to the problem, and thus have to change your calculation. There's a 1/20 chance of rolling 2 1s after you're already rolled one. Let's calculate it... So, there's 400 ways 2 dice can fall, yes, and there's only 1 way that they can both fall on 1. However, there's 20 ways that the first die can fall on 1, one for each possible fall of the second die. So, when we say that that has already happened, we have to eliminate 380 of those 400 die rolls, those are no longer possible. That leaves us with only 20 ways that the second die can fall, and only 1 of those is a 1. So the odds of rolling a on the second die, after already rolling a 1 on the first die is 1/20. We can also calculate it differently. What are the odds of the second die falling on 1? Cause that's the one we care about, really. And there's 20 ways that can happen, one for each possible fall of the first die. So the odds of the second die falling on 1, when rolling 2 dice is 20/400, or 1/20.

D4 is the devil's dice.

Imagine if he didn't always show the other goat. "So you picked door number one. Let's see what's behind door number 2; a brand new car! ...so, do you wanna switch to door 3?"

These dice are spinning around me The whole table's spinning without me Every sesh sends future to past Every roll leaves me one less to my last

I don't know but if you rub my lucky clover you'll get a little squirt of luck.

Maybe the real Cthulhu was the impossibly mind-breaking irrational thought experiments we subjected ourselves to along the way!

Short version, two coin flips. There are 4 options: HH, HT, TH, TT So there's two chances to get one Tails and one Heads, out of 4, so 2/4 = 1/2, half the tosses. Then 1/4 on each of HH and TT. So rolling one Tails is more likely than rolling two. But once you've flipped the first coin, it's "locked in". If it was Heads, the only options left to you are HT and HH. The fact that there *could* have been a T that, if flipped first, would land us in TH is irrelevant fantasy. We've got the H, and all that's left is HT or HH, even odds. Dice are the same. What makes a double 1 rare is that you have to roll 1 specifically and only two times to get there, whereas a single 1 can be first or second, and the other number can be any of the other 19 other numbers. It's the duplication of different results we consider "the same" that make one thing more likely. But once you've already rolled a 1, none of that matters anymore. Now it's just 20 numbers, each equally likely. We're locked in.

Roll my number, roll my number, roll my number, I'm not afraid...

I think the problem is that people forget *Monty Hall* has information that the contestant does not. The naive assumption is that he's just picking a door and you're just picking a door. The unsophisticated viewer never really stops to think about why Monty Hall never points to a door and reveals a prize by mistake. One way I've had success explaining it is to expand the problem to more than three doors. Assume 100 doors. Monty Hall then says "Open 98 doors" and fails to reveal a prize behind any of them. Now its a bit more clear that he knows something you don't.

The same logic applies to a nat 20 though

Maybe? I don't think that was my issue. I think I was overthinking it and using the second "choice" as an event with separate odds.

And of course the traditional sentence for dice which misbehave one too many times. 

The thing you're getting by switching is the benefit of the information provided by the person who revealed an empty door. Before a door is open, you have a 1/3 chance of selecting correctly. After you select a door, the host picks from the other two doors. This provides extra information you didn't have during your initial selection. The host points to a door *they know is a dud* and asks for it to open. So now you're left with the question "Did I pick the correct door on the first go? Or did the host skip the door that had the prize?" There's a 1/3 chance you picked the right door initially and a 2/3 chance the host had to avoid the prize-door.

Where did they get an AI that managed to mess up "roll" as "role" twice in the same page? Humans do it because they sound the same, but AI doesn't know how they sound. The AI knows that sometimes people say "role" instead of "roll", but they're generally set to raise the probability of a token to some power, and since most people spell "roll" right, they're even more likely to. And they also generally have a post-training step where they're trained to spell stuff right and that sort of thing. And they don't even need to be trained on that specifically, since some people spell better than others, so they can understand the general concept of good vs bad spelling.

That’s stupid. But obviously how the dice strikes the table impacts its balance and therefore the probability of rolling specific numbers. So we must figure out what side need to strike the table first to decrease the probability of getting an undesirable roll. Boom, I out physicsed you’re probabilities.

The funny thing is that this logic assumes the rolls are independent (so you can just multiply probabilities), but the definition of independence is that past rolls can't affect future ones. So basically it's saying that past rolls can't affect future ones and therefore they must.

Me every time I think about this. 

The math checks out, but the problem is the danger of rolling a nat 20 on your practice roll. The odds of getting two nat 20s in a row are almost as low as the odds of getting two nat 1s, so you may be screwing yourself out of a crit

Weirdly enough, it’s just the way probability works. Once something stops being a possibility, and becomes a fact (ie. dice are rolled, numbers known) - future probability is no longer affected (assuming independent events like die rolls). e.g. you have a 1/400 chance of rolling two 1s on a D20 back-to-back. But if your first roll is a 1, you’re back down to the standard 1/20 chance of doing it again - because one of the conditions has already been met.