Your list actually doesn’t go far enough. There is a fourth, and scarier category. Things which would, if possibly render probability useless as a model. “The chance that probabilities don’t apply to anything.” is in the fourth category. I would also place anything that violates such basic things as the consistency of physics, or the existence of the external world.
For really small probabilities, we have to take into account some sources of error that just aren’t meaningful in more normal odds.
For instance, if I shuffle and draw one card from a new deck what is the chance of drawing the ace of spades? I disregard any chance of the deck being defective, any chance of my model of the universe being wrong, and any chance of laws of identity being violated. Any probabilities are eclipsed by the normal probabilities of drawing cards. (category 1)
If I shuffle and draw two cards without replacement from a new deck, what is the chance of them both being aces of spades? Now I have to consider other sources of error. There could have been a factory error or the deck may have been tampered with. (category 2)
If I shuffle and draw one card from a new deck, what is the chance of it being a live tiger? Now I have to consider my model of the universe being drastically wrong. (category 3)
If I shuffle and draw one card from a new deck, what is the chance of it being both the ace of spades and the two of clubs? Not a misprint, and not two cards, but somehow both at the same time. Now I have to consider the law of identity being violated. (category 4)
It would more likely be user error. I believe 53 is prime. If it isn’t then either mathematics is broken or I have messed up in my reasoning. It is much more likely that I made an error or accepted a bad argument.
53 not being prime while having no integer factors other than 1 and itself would break mathematics.
Your list actually doesn’t go far enough. There is a fourth, and scarier category. Things which would, if possibly render probability useless as a model. “The chance that probabilities don’t apply to anything.” is in the fourth category. I would also place anything that violates such basic things as the consistency of physics, or the existence of the external world.
For really small probabilities, we have to take into account some sources of error that just aren’t meaningful in more normal odds.
For instance, if I shuffle and draw one card from a new deck what is the chance of drawing the ace of spades? I disregard any chance of the deck being defective, any chance of my model of the universe being wrong, and any chance of laws of identity being violated. Any probabilities are eclipsed by the normal probabilities of drawing cards. (category 1)
If I shuffle and draw two cards without replacement from a new deck, what is the chance of them both being aces of spades? Now I have to consider other sources of error. There could have been a factory error or the deck may have been tampered with. (category 2)
If I shuffle and draw one card from a new deck, what is the chance of it being a live tiger? Now I have to consider my model of the universe being drastically wrong. (category 3)
If I shuffle and draw one card from a new deck, what is the chance of it being both the ace of spades and the two of clubs? Not a misprint, and not two cards, but somehow both at the same time. Now I have to consider the law of identity being violated. (category 4)
Would 53 not being prime break mathematics?
It would more likely be user error. I believe 53 is prime. If it isn’t then either mathematics is broken or I have messed up in my reasoning. It is much more likely that I made an error or accepted a bad argument.
53 not being prime while having no integer factors other than 1 and itself would break mathematics.
LNC, not the law of identity, I think.
Oops, right. Non-contradiction.