What is the probability that $n$ persons have birthdays all within one half year?

Question

I'm wondering what the probability is that $n$ persons have birthdays all within 1/2 year. For instance, two persons always have birthdays within 1/2 year, as the maximal distance between two days is at most 1/2 year. I suspect this problem to be related to the broken stick problem, which predicts the probablity of $n-1$ points to all lie on 1/2 of a circle to be $$ \mathbb P = \frac{n}{2^{n-1}} $$ For my example above, $n-1=2$ gives $\mathbb P=\frac{3}{4}$, so, clearly, I misunderstand something here.