There are $k$ rooms with $n$ lights each. Each light is on with equal probability $p$ independently of other lights. As $k$ stays fixed and $n$ goes to infinity, what is the limit of the probability that Room 1 has the maximum number of lights on (possibly sharing this maximum with other rooms)?
By symmetry, this probability is clearly at least $1/k$ for any $n$. However, for fixed $n$ it is slightly larger than $1/k$ because the maximum can be equal for many rooms. Still I think the limit should be $1/k$.