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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.

RobPratt
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adriaanJ
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    I think the answer to the linked question deals with $n$ points (not $n-1$). – drhab Dec 28 '23 at 10:25
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    The part about "$n-1$ points" in the accepted answer is a step that is taken to transform an answer about $n$ points on a ring to one about $n-1$ points on a straight stick. But you are asking about a ring (because you don't consider December to be more than half a year from January), so you should ignore the part of the answer that talks about straight sticks. You just want the part that talks about $n$ points on a ring. So $3$ points leads to probability $\frac34$, but $2$ points leads to probability $1$. – David K Dec 28 '23 at 15:45

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