# 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 a
ll.

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




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

0.

It is easy to explain since (Finite) / (Infinite) = 0 anyway. More
over, the probability of the number being a rational number is sti
ll 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 uni
on 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 p
robability of the number belonging to the set is still 0. The leng
th (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 probabilit
y. Also those are far more geometric than finite cases.