# The Probability Is 0

POSSIBILITY DOES NOT MEAN PROBABILITY.

I think this is a common sense for people who have studied math in
university. But surprisingly general population don't know it at
all.

I don't know how to explain this more easily. So if you don't
understand the terms please search them.




Suppose you pick a real number between 0 and 1. What is the
probability of the number being 1?

0.

It is easy to explain since (Finite) / (Infinite) = 0 anyway.
Moreover, the probability of the number being a rational number is
still 0. Also easy to explain. (Countable) / (Uncountable) = 0.

In fact, what determines a probability is the length of a set. Or
area, volume, &c. We call it "Measure" in general.


The set {1} is nonempty. But the length of it is 0. A point has no
length. Thus the probability is 0.

Similarily, although the set of all rational numbers between 0 and
1 is infinite, the probability is still 0. Since the countable
union of measure zero sets is still measure zero. It is the countable
union of each point ({1}, {1/2}, &c.) which is measure zero.

More unintuitively, although Cantor set is even uncountable, the
probability of the number belonging to the set is still 0. The
length (based on Lebesgue measure) is 0.

However, the probability of the number being between 0 and 1/2 is
1/2. Since the length of [0,1/2] is 1/2.


Therefore for infinite cases, possibility does not mean probability.
Also those are far more geometric than finite cases.