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I found the following link on youtube.com:

Probability of zero is not impossible

$\text{Isn't } \textbf{P(0)} = 0 \;\;?$

I did not understand this video. Can anyone kindly tell me why that is so? Why probability of zero is not impossible?

I am asking the above question because in the following link at the bottom of the website, one of the poster said the probability of zero means impossibility. Then? Where is my error in understanding? I am really confused now. Can anyone kindly elaborate on this paradoxical math problem or is it really a paradox?

Probability of zero means impossible

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    Most mathematical "paradoxes", including this one, rest on poorly (if at all) defined terms. What do you mean with "impossible"? – Git Gud Apr 15 '20 at 23:21
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    The five-second version: choose a real number uniformly at random between 0 and 1. The (a priori) probability of choosing any given number is zero, but that even holds true for whatever number gets chosen. – Steven Stadnicki Apr 15 '20 at 23:21
  • @GitGud That the possibility of the event to occur is zero. –  Apr 15 '20 at 23:24
  • What do you mean with "possibility"? – Git Gud Apr 15 '20 at 23:25
  • @GitGud There are two outcomes to an event. Either it will occur or it will not. Shouldn't Probability of zero means it won't happen at all? –  Apr 15 '20 at 23:35
  • @GEdgar I got the answer to my question answered below. By peter.petrov –  Apr 16 '20 at 00:03

1 Answers1

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This usually pertains to uncountable infinite sets/spaces.

Say you have all the points in the closed unit circle
(in the Euclidean plane) and you pick a point at random.

1) What is the probability you would pick the point (0,0) ?

2) What is the probability you would pick a point from the segment OB, where O=(0,0), B=(1,0) ?

Both probabilities are 0. But both are possible.

Why are both zero? You need to read a bit about Lebesgue measure.

The first few chapters here are very nice and not too heavy and they cover Lebesgue measure, and all the basics/foundations on which modern probability theory is based (I mean based as a serious axiomatized math discipline, not just as an empirical study /which it initially was/).

Harald Cramér - Mathematical Methods in Statistics

peter.petrov
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