Picking a real number from the unit interval with uniform distribution is having Khinchin's constant with probability $1$ as the geometric mean of its continued fraction's coefficients, not considering the first $0$.
On the other hand, if you generate a random real number by picking random coefficients for the continued fraction you will end up with a number with a property that it's not having Khinchin's constant as GM with probability $1$. As I can't pin down a distribution for picking infinite naturals let's say you can choose from 0-100 uniformly for coefficient the statement of not getting the Khinchin's property is still true.
In the first method, you have $2^{\aleph_0}$ cardinality the second method you have ℵ0^ℵ0 cardinality (well 101^ℵ0 in case of restricting it).
Are there really more reals in the second case? Where does my reasoning fail?
I have a feeling that it all comes down to the definition of 'choosing randomly' and representations of reals.
** I have a doubt that this statement is true at all.