Random Room changing in the Hilbert hotel.

Question

Let's say you have a Hilbert's grand hotel full occupancy. Assign each occupant a new room select randomly without regard to whether the room is assigned to someone. i.e. empty rooms, multiple occupancies etc are all allowed.

what are the chances that any given room is vacant?

I get $1/e$ the limit of $((x-1)/x)^x$ as x -> ∞.

but

if each member of an infinite set has a > 0 chance of doing something then that thing should happen and happen an infinite number of times making a 0 chance of having an empty room or even a room with < infinite members.

So what gives?

Answer 1

The limit you took doesn't correspond to any limiting distribution. For each $x$, each occupant is assigned one of the rooms with the same probability $\frac1x$. But infinitely many occupants cannot be assigned rooms with the same probability "$\frac1\infty$".

Here's a distribution for the infinite case, though: Inhabitant $n$ takes one of the rooms $1$ through $n$ with equal probability $\frac1n$. Then room $k$ remains empty with probability

$$ \prod_{n=k}^\infty\left(1-\frac1n\right)\;. $$

This infinite product diverges to zero, so all rooms are almost surely occupied.

If you use the same probabilities in the finite case with $x$ occupants, then for fixed $k$ the probability for room $k$ to remain empty will go to $0$ as $x\to\infty$.

Answer 2

Related to joriki's answer,
if Inhabitant $n$ goes to a room between $1$ and $n\log n$, each room is also almost surely occupied, but if they go to a room between $1$ and $n^2$, the probability is less than $1$.