I would assume that the chance would be $0%$ as $0=\lim\frac{1}{n}$ but since the equivalent would be something along the lines of $\frac{\aleph_0}{\aleph_1}$ I am not sure.
If $0=\frac{\aleph_0}{\aleph_1}$ would that not mean that anything would be an infinite subset when randomly sampled from an uncountable set, even elements that are not in the set with uncountable cardinality to begin with?
There is probably some statistics and cardinality knowledge that i am missing but hopefully someone can give me some understanding :O