Is there a situation where we know that 'random' objects have a given property but we still have no example of an object with said property?
For example, someone could use Cantor's diagonal argument to show that real numbers were uncountable even though algebraic numbers are countable. So with the standard probability measure on $[0,1]$, random real numbers will be transcendental. But it requires a very different argument to give/construct an example of a transcendental number.
So my question is whether we can replace 'transcendental' with some other property for which we know that almost all real numbers have that property but we have no explicit example of such a real number.
And since I'm talking about random objects, you can replace 'real numbers' with other sample spaces.
For another example, it has been shown that 'random' groups have many surface subgroups (see here) and we actually have explicit examples of groups with many surface subgroups: the group of isometries of the hyperbolic plane.
Can 'has many surface subgroups' be substituted with another property to give a true statement on random groups but for which we have no explicit example of a group with that property?
I was watching the latest video from PBS Infinite Series and she mentions normal numbers towards the end. These are example of what I was asking for.
It is known that almost all real numbers are (absolutely) normal and I saw a proof in Ergodic Theory last semester. This article gives a partial proof since it uses a deep theorem from ergodic theory without proof. What I had forgotten was the discussion we had in class about explicit examples. Wikipedia lists examples of numbers that are normal with respect to particular bases but none that are absolutely normal, i.e., normal in all bases.