If you pick up and roll a d20, you have a 1 in 20 chance of rolling any given number. If you pick it up again and roll it, you have a 1 in 20 change of getting the same number despite the fact that, if you’d considered this from the outset, you’d have had a 1 in 400 change of rolling the same number. If you rolled it a third time, it’s still 1 in 20 as roll it but, as a concept, is also 1 in 8000.
Mathematically speaking, the 1 in 8000 chance represents all potential outcomes for rolling a twenty-sided die three times in succession with the goal of it being the same number each time, but technically only if you pick which number it is in advance. If you say “any number” three times, the chance is really 1 in 400 because the first roll just sets the target number rather than affecting the probability of getting it three times in a row. But, if you talk about it afterwords, you still had a 1 in 8000 chance of rolling that exact sequence.
Clearly, the probability of this is a bit fucky because anyone who has played D&D can tell you, you do not need to roll a d20 8000 times in order to roll three natural twenties in a row. In 5e, this exact sequence is a little less remarked upon because if you roll a single “natural” 20, you critically succeed at whatever the roll was about (some variations on this apply since that’s technically a house rule that is widely used rather than a rule suggested by the D&D books). You don’t need to roll anything else. In the previous popular version of D&D, 3.5, you had to roll to confirm your critical hit and almost everyone played with the custom rule that rolling two 20s in a row mean you went from scoring a critical hit to potentially killing your target with a single hit. If you rolled a third 20, your target was just dead because it had been made clear the dice gods wanted them dead.
As a DM in charge of NPCs in combat and in social encounters, I typically make more dice rolls than any one of my players. As a result I have a tendency to see more extremes than most of my players. That being said, it doesn’t account for my ability to roll three natural twenties as frequently as I do. I can count on one hand the number of times I’ve roll three natural ones in a roll because the number is zero. The most I’ve ever rolled is two. Yet I’ve also rolled five natural twenties in a row. I also have a high tendency to roll in the upper third of numbers on any standard die I roll. I like to joke with my friends that prettier dice roll better, but it doesn’t matter for me. Any die I pick up can do this.
I can’t explain it any way but this: I have a huge amount of dice luck. It doesn’t apply to any kind of chance-oriented situation since I’ve only ever won one thing in a drawing or lottery, and I don’t think I’ve ever been dealt a decent hand in any card game, but I always roll high with dice. Specifically, high. Not “well.” My friend was working on a game where you needed to roll under certain numbers in order to succeed and I failed literally every check I ever rolled in the two 3-hour sessions we played that game. For whatever reason, I roll high on dice.
Which is an incredibly frustrating knack to have as a DM. I’ve killed more players due to my own good dice rolls than because of my players’ poor roles or bad decisions. As a player in other people’s games, it can be frustrating or seem like cheating when I succeed with aplomb more frequently than the rest of the table put together. I’ve been called on it a dozen or so times and now I just roll my dice in the open as a player. If they call me on it then, I offer to re-roll with anyone’s dice and still, I succeed. This is one of the reasons I always get nervous about the idea of playing in a game shop. It only takes one person losing their temper and accusing you of cheating to make you not want to play with strangers any more.
So when I roll three twenties in a row and insta-kill two player characters, I sigh heavily and move on with my life. As do my players. It’s not like I tried. I didn’t choose to be an outlier on the bellcurve that is the laws of probability.
I actually took a statistics class in college, back when I was studying psychology, and I have to say that most statistics and most probability is a bunch of bullshit when it comes to explaining how the world works. Statistics are helpful because they are part of understanding how the world works on a conceptual level, but all you have to do is look into the Monty Hall Problem to realize it is a difficult science with far more factors than is readily apparent which makes it far too complex for most of us to understand without a great deal of explanation.
Anyway, the real purpose of this piece is so that I have something I can link to when my players complain about how frequently I roll natural 20s in our D&D games and so other people can appreciate that yes, sometimes your luck really is just that good/bad. You have no one to blame for your dice rolls but yourself since not even the uncaring mathematical universe controls them as we can clearly see from the fact that I just dumped out my bag of d20s at my desk and had five twenties out of the 27 dice I rolled.