You seem to think probabilities of 0 and 1 are mysterious or contradictory when discussing randomness; they aren't. When you're talking about randomness, you need to define your support. that mere action gives you places where the probability is zero. For example: Can the time to run 100m ever be negative? No? Then P(t<0) = 0. And by extension, P(T>=0) = 1.

No puzzle there. But you're transfrormation to log-odds has some regularity conditions you're violating in those cases: the transform is only defined for probabilities in (0,1). But that doesn't mean log-odds or probabilities are flawed. Probabilities or 0 and 1 -- like log-odds of plus-and-minus infinity -- are just filling in the boundaries on the system you've created. Mathematically, you want to be able to handle limits; that means handling limits as a probability approaches 0 or 1. That's it.

This shouldn't be some huge philosophical puzzle; it's merely the need to have any mathematical system you use be complete. Sir David Cox would be the first to tell you that.

Comment author:[deleted]
24 May 2013 06:22:25AM
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We certainly can talk about the limit of a function whose codomain is a measure of probability being 1; the limit of the probability of a proposition as the amount of evidence in favor of it approaches infinity is 1. But that doesn't mean that 1 is a measure of probability. Infinity is valid as the limit of a function yielding real numbers, but infinity is not a real number.

As for your example with the amount of time it takes to run a particular distance, I can't be certain that we won't find a region of space with strange temporal effects that allow you to take a walk and arrive at your starting point before you left. This would allow you to run a hundred meters in negative time, in at least one sense of the word. Getting that sort of speed from the runner's point of view would be stranger, but the Dark Lords of the Matrix could probably make it happen.

## Comments (128)

OldYou seem to think probabilities of 0 and 1 are mysterious or contradictory when discussing randomness; they aren't. When you're talking about randomness, you need to define your support. that mere action gives you places where the probability is zero. For example: Can the time to run 100m ever be negative? No? Then P(t<0) = 0. And by extension, P(T>=0) = 1.

No puzzle there. But you're transfrormation to log-odds has some regularity conditions you're violating in those cases: the transform is only defined for probabilities in (0,1). But that doesn't mean log-odds or probabilities are flawed. Probabilities or 0 and 1 -- like log-odds of plus-and-minus infinity -- are just filling in the boundaries on the system you've created. Mathematically, you want to be able to handle limits; that means handling limits as a probability approaches 0 or 1. That's it.

This shouldn't be some huge philosophical puzzle; it's merely the need to have any mathematical system you use be complete. Sir David Cox would be the first to tell you that.

We certainly can talk about the limit of a function whose codomain is a measure of probability being 1; the limit of the probability of a proposition as the amount of evidence in favor of it approaches infinity is 1. But that doesn't mean that 1 is a measure of probability. Infinity is valid as the limit of a function yielding real numbers, but infinity is not a real number.

As for your example with the amount of time it takes to run a particular distance, I can't be certain that we won't find a region of space with strange temporal effects that allow you to take a walk and arrive at your starting point before you left. This would allow you to run a hundred meters in negative time, in at least one sense of the word. Getting that sort of speed from the runner's point of view would be stranger, but the Dark Lords of the Matrix could probably make it happen.