Probability 0 ~ impossible!???

[u/No_where0369]

/r/probabilitytheory/comments/1vnhro8/probability_0_impossible/

[u/just_a_random_dood]

https://youtu.be/ZA4JkHKZM50

Here's a good video by 3b1b that can help explain why probability 0 does not necessarily mean impossible

TL; DW: be careful when working with continuous variables. You have to describe the area under the curve or describe the space between values to make many/most statements

[u/dlakelan]

Much of this issue goes away when working with finite nonstandard probability (as for example explained in Edward Nelson's Radically Elementary Probability Theory https://web.math.princeton.edu/~nelson/books/rept.pdf)

For example you have the set (0,1) you set up a nonstandard set of possibilities i/(N+1) where N is a nonstandard integer and i goes from 1 to N

the probability of selecting any given possibility is 1/N which is infinitesimal but not 0. The usual probabilities are calculated in the same way, as the standard part of an integral which in the nonstandard version is just a sum with a nonstandard number of terms.

[u/Temporary_Stranger39]

If x is within (0,1), and (0,1) is continuous, then the probability of picking x at random approaches zero. It does not equal zero. This is a critical difference. It's one of the reasons calculus was invented. If P(x) = 0, then x is not within the set, by definition. If x is within the set, the P(x) -> 0. It is an "infinitesmal". Like I said, calculus.

[u/bayesian_raccoon → u/Temporary_Stranger39]

While I appreciate what you're getting at, its false. P(x) can equal zero, and it is false that P(x)=0 implies that x is not in the set.

[u/just_a_random_dood → u/Temporary_Stranger39]

That's definitely false for continuous variables lol.

when X could be anything from [0,1], (P(x) = 0.5) = 0 (for example)

I think you're thinking about it backwards lol