This is not quite accurate. You can’t uniformly pick a random rational number from 0 to 1, because there are countably many such numbers, and any probability distribution you assign will have to add up to 1. Every probability distribution on this set assigns a nonzero probability to every number.
You can have a uniform distribution on an uncountable set, such as the real numbers between 0 and 1, but since you can’t pick an arbitrary element of an uncountable set in the real world this is theoretical rather than a real-world issue.
As far as I know, any mathematical case in which something with probability 0 can happen does not actually occur in the real world in a way that we can observe.
As far as I know, any mathematical case in which something with probability 0 can happen does not actually occur in the real world in a way that we can observe.
Here’s one example:
What’s the probability that our physical constants are what they are, especially the constants that seem tuned to life?
The answer is if the constants are arbitrary real numbers, the answer is probability 0, and this applies no matter what number you pick.
This is how we can defuse the fine-tuning argument, that the cosmos’s constants have improbable values that seem tuned for life, since any other constant has probability 0, no matter whether it was able to sustain life or not:
The question to ask is, what is the measure of the space of physical constants compatible with life? Although that requires some prior probability measure on the space of all hypothetical values of the constants. But the constants are mostly real numbers, and there is no uniform distribution on the reals.
Thanks, I didn’t realize that! It does make sense now that I think about it. I think if you replace the rationals with the reals in my theoretical example, the rest still works?
And yes, I agree about in the real world. Probabilities 0 and 1 are limits you can approach, but only reach in theory.
This is not quite accurate. You can’t uniformly pick a random rational number from 0 to 1, because there are countably many such numbers, and any probability distribution you assign will have to add up to 1. Every probability distribution on this set assigns a nonzero probability to every number.
You can have a uniform distribution on an uncountable set, such as the real numbers between 0 and 1, but since you can’t pick an arbitrary element of an uncountable set in the real world this is theoretical rather than a real-world issue.
As far as I know, any mathematical case in which something with probability 0 can happen does not actually occur in the real world in a way that we can observe.
Here’s one example:
What’s the probability that our physical constants are what they are, especially the constants that seem tuned to life?
The answer is if the constants are arbitrary real numbers, the answer is probability 0, and this applies no matter what number you pick.
This is how we can defuse the fine-tuning argument, that the cosmos’s constants have improbable values that seem tuned for life, since any other constant has probability 0, no matter whether it was able to sustain life or not:
https://en.wikipedia.org/wiki/Fine-tuned_universe
The question to ask is, what is the measure of the space of physical constants compatible with life? Although that requires some prior probability measure on the space of all hypothetical values of the constants. But the constants are mostly real numbers, and there is no uniform distribution on the reals.
Thanks, I didn’t realize that! It does make sense now that I think about it. I think if you replace the rationals with the reals in my theoretical example, the rest still works?
And yes, I agree about in the real world. Probabilities 0 and 1 are limits you can approach, but only reach in theory.