I mean, regardless of what were the previous two rolls—let’s call them “X” and “Y”—if the next roll comes up 20, we are witnessing a sequence “X, Y, 20”, which has a probability 0.0125%. That’s true even when “X” and “Y” are different than 20.
You could make the sequence even more improbable by saying “if this next roll comes up, we are witnessing an extremely improbably sequence—we are living in a universe whose conditions allow creation of matter, we happen to be on a planet where life exists, dinousaurs were killed by a comet, I decided to roll the 20-sided-die three times, the first two rolls were 20… and now the third roll is also 20? Well this all just seems very very unlikely.”
Or you could decide that the past is fixed, if you happen to be in some branch of the universe you are already there, and you are only going to estimate the probability of future events.
Even better, what ChristianKl said. A better model would be that depending on the existing state of the house there is a probability P saying how frequently things will break. At the beginning there is some prior distribution of P, but when things start breaking too fast, you should update that P is probably greater than you originally thought… and now you should expect things to break faster than you expected originally.
regardless of what were the previous two rolls—let’s call them “X” and “Y”—if the next roll comes up 20, we are witnessing a sequence “X, Y, 20”, which has a probability 0.0125%. That’s true even when “X” and “Y” are different than 20.
Yes, all sequences X,Y,Z are equally (im)probable if the d20 is a fair one. But some sequences—in particular those with X=Y=Z, and in more-particular those with X=Y=Z=1 or X=Y=Z=20, are more likely if the die is unfair because they’re relatively easy and/or relatively useful/amusing for a die-fixer to induce.
As you consider longer and longer sequences 20,20,20,… their probability conditional on a fair d20 goes down rapidly, whereas their probability conditional on a dishonest d20 goes down much less rapidly because there’s some nonzero chance that someone’s made a d20 that almost always rolls 20s.
The “however” part seems irrelevant.
I mean, regardless of what were the previous two rolls—let’s call them “X” and “Y”—if the next roll comes up 20, we are witnessing a sequence “X, Y, 20”, which has a probability 0.0125%. That’s true even when “X” and “Y” are different than 20.
You could make the sequence even more improbable by saying “if this next roll comes up, we are witnessing an extremely improbably sequence—we are living in a universe whose conditions allow creation of matter, we happen to be on a planet where life exists, dinousaurs were killed by a comet, I decided to roll the 20-sided-die three times, the first two rolls were 20… and now the third roll is also 20? Well this all just seems very very unlikely.”
Or you could decide that the past is fixed, if you happen to be in some branch of the universe you are already there, and you are only going to estimate the probability of future events.
Even better, what ChristianKl said. A better model would be that depending on the existing state of the house there is a probability P saying how frequently things will break. At the beginning there is some prior distribution of P, but when things start breaking too fast, you should update that P is probably greater than you originally thought… and now you should expect things to break faster than you expected originally.
Yes, all sequences X,Y,Z are equally (im)probable if the d20 is a fair one. But some sequences—in particular those with X=Y=Z, and in more-particular those with X=Y=Z=1 or X=Y=Z=20, are more likely if the die is unfair because they’re relatively easy and/or relatively useful/amusing for a die-fixer to induce.
As you consider longer and longer sequences 20,20,20,… their probability conditional on a fair d20 goes down rapidly, whereas their probability conditional on a dishonest d20 goes down much less rapidly because there’s some nonzero chance that someone’s made a d20 that almost always rolls 20s.