[baduser]

> Probability-zero events happen all the time. The probability of getting any specific value selected uniformly at random from the unit interval (say, 0.232829) is zero.

I would strongly challenge that claim. First, you did not choose that number uniformly at random, you chose it from at best a countably infinite subset, or more realistically, from a finite subset. And secondly, I do not think you can describe a situation where a number is actually chosen uniformly and randomly from the unity interval.

[rbanffy]

But, if the set is infinite, won’t the odds of any individual number to be selected be effectively zero?

[998244353]

It is, and if you choose the number uniformly at random, it's not just "effectively" zero, it is precisely zero.

GP's point, as I understand it, is that it is not actually possible to choose a number from [0, 1] uniformly at random in "real life".

I think you could argue that, e.g. in the dartboard thought experiment, the probability of choosing individual points doesn't really matter: only probabilities of measurable subsets with positive measure matter.

[baduser]

I guess, but, the set is not even countably infinite. "Selecting at random" is something that happens in the real, non-infinite world, not in the mathematically rigorous would where infinities can exist. So, no, probability-zero events do not happen in either.

[rbanffy]

Not necessarily - you might just come up with a number you know is in the set, say pi/4. I know it's in the set because it satisfies the conditions that define it. Still, the odds of that particular number being picked up are zero.

[baduser]

If you picked that, you did not pick uniformly from the unit interval. You picked from something else, which was much smaller.

[bryanrasmussen]

so how does the random function selecting the number select the number before the last number in the infinite set?

[rbanffy]

It wouldn't "select" as if it were an infinite deck of cards, but rather generate a number we know is on the infinite set. It can very well take an infinite amount of time to come up with the digits though...

[baduser]

> rather generate a number

How?

[rbanffy]

A series of random bits?

[shiandow]

What last number?

[bryanrasmussen]

exactly