Congratulations to August’s winner, Jelmer H., who solved the problem by finding and inverting the cumulative distribution function for the number of nurses.
“There’s always a bug going round, so it’s important to keep hospitals well staffed. Government regulations require that each hospital maintain a ratio of at least 1 nurse to every 2 patients, 99% of the time.
If City A has 5,000 people and City B has 50,000, what is the ratio of nurses at hospital A to those at hospital B?”
on first read, it may seem like Hospital A, serving a population one tenth the size of Hospital B’s, would need a tenth as many nurses and doctors.
and, if we were thinking about average outcomes, that would be true.
let’s say the probability that a member of the population requires hospitalization in a given day is $p$, then the number of patients we’d expect to show up at City Hospital A in a day is $N_A\ p$ where $N_A$ is the population of City A. likewise, City Hospital B expects $N_B\ p$ patients so the ratio of patients is the ratio of the populations.
however, events are less predictable in small settings than large ones. we can see this by think about an extreme case.
if $p=0.1$, and the city’s population is $10$ then we’d expect $1$ patient per day, though it would not be surprising if the number on a given day were $0$ or $2$.
by contrast, in a city of population of $1,000,000$ we expect $100,000$ patients per day, but it would be shocking if any given day yielded $0$ or $200,000$.
the upshot is that large fluctuations relative to the mean are more likely in small cities than in big ones.
how can we quantify this?
to comply with the regulation, each hospital needs to maintain a 1:2 nurse-to-patient ratio 99% of the time.
we can’t ensure this by thinking about the typical outcome — we have to think about how the number of daily patient admissions is distributed.
according to the assumptions of the question, each member of the population has probability $p = 0.0015$ to be a patient in a hospital, per day. so, the probability for any particular group of $N$ people to have $a$ admits among them is $p^a(1-p)^{N-a}$.
we also have to account for how many ways there are to choose $a$ people to admit from a population of $N$. this is the binomial coefficient, $\binom{N}{a} = \frac{N!}{(N-a)!a!}.$
putting these together, the probability for the city hospital to admit $a$ patients in a day is then
$$ P(a)=\binom{N}{a}p^a(1-p)^{N-a}. $$