Probability à la Tristram Shandy

Hilbert's Hotel! A set with countably infinite number of elements can accommodate finitely many new members! The probability of picking any apple is $1$ and the probability of picking ALL apples is also $1.$ In a way an intuitive connection to Zero-One Laws.

In more detail: the probability of picking one given apple in $N$ trials is

$\displaystyle P(N)=\sum_{cycl}\frac{\displaystyle \left(1-\frac{1}{9i+1}\right)^i}{9i+1}.$

We can show that $\displaystyle \lim_{N\to\infty}P(N)=1.$

The problem has been discussed previously two decades ago. It reflects on the Tristram Shandy paradox by Bertrand Russell.