3. ELI5: Probability that a random real number from the interval 0 to 1 is irration...

ELI5: Probability that a random real number from the interval 0 to 1 is irrational.

Probability you pick a rational number is (number of rational numbers) / (number of real numbers)
So a rational number is any number that can be represented as something like x / y, where x and y is a whole number.
Now you've probably been told that eventually no matter how complicated it is, a rational number will eventually start repeating.
So I would like you to imagine that some one put all the ways a sequence of numbers can repeat onto a big huge table of numbers.
So stuff like 000000000000000000000000000000000000000000000000000000...
111111111111111111111111111111111111111111111111111111...
101010101010101010101010101010101010101010101010101010...
110110110110110110110110110110110110110110110110110110...
Continuing infinitely in all directions.

Now consider this, if you were to take a diagnoal sample of this table, EG: 000000000000000000000000000000000000000000000000000000...
111111111111111111111111111111111111111111111111111111...
101010101010101010101010101010101010101010101010101010...
110110110110110110110110110110110110110110110110110110...
and treat that as a number, would it repeat?
No, it won't since the sequences never repeat.
This number is an irrational number, since it doesn't repeat.
Now shift the number one to the left, is that the same number?
No.
How many times can we shift it?
Infinite times since it's infinitely long.
So that means there are infinitely more irrational numbers than there are rational numbers.
This is called Cantor's Diagonal Argument.
There are much more than infinite numbers of real numbers than there are rational numbers.
So your odds are less than 1/infinity, which is zero.

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