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 -> ∞.
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?