I was saying things like, “I really doubt the garbage disposal is really broken, we just had two other major things replaced, what are the odds that another thing would break so quickly?” [...] I am probably a poster child for “doing probabilistic thinking wrong” in some obvious way that I am blind to.
You are indeed doing it very wrong. As far as proablisitic reasoning goes the fact that one item broke doesn’t reduce the chances that a second item breaks at all.
Yeah, okay, I worded that stupidly. It’s more like this:
“This 20-sided-die just came up 20 twice in a row. The odds of three consecutive rolls of 20 is 0.0125%. I acknowledge that this next roll has a 1⁄20 chance of coming up 20, assuming the die is fair. However, if this next roll comes up 20, we are witnessing an extremely improbable sequence, so improbable that I have to start considering that the die is loaded.”
However, if this next roll comes up 20, we are witnessing an extremely improbable sequence, so improbable that I have to start considering that the die is loaded.”
The equivalent of “considering that the die is loaded” in your example is “the previous owners did a bad job of maintaining the house”. It’s indeed makes sense to come to that conclusion. That’s also basically what your wife did.
Apart from that the difference between sequences picked by humans to look random and real random data is that real random data more frequently contains such improbable sequences.
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.
You are indeed doing it very wrong. As far as proablisitic reasoning goes the fact that one item broke doesn’t reduce the chances that a second item breaks at all.
Yeah, okay, I worded that stupidly. It’s more like this:
“This 20-sided-die just came up 20 twice in a row. The odds of three consecutive rolls of 20 is 0.0125%. I acknowledge that this next roll has a 1⁄20 chance of coming up 20, assuming the die is fair. However, if this next roll comes up 20, we are witnessing an extremely improbable sequence, so improbable that I have to start considering that the die is loaded.”
The equivalent of “considering that the die is loaded” in your example is “the previous owners did a bad job of maintaining the house”. It’s indeed makes sense to come to that conclusion. That’s also basically what your wife did.
Apart from that the difference between sequences picked by humans to look random and real random data is that real random data more frequently contains such improbable sequences.
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.