= Matching one specified birthday
{title2=$1-(1-1/m)^{n-1}$}
In a group of total size $n\geq1$, designate one member and ask whether any of the other $n-1$ labels equals that member's label. Conditional on the designated label, all the other draws avoid it with probability $1-1/m$. <Independence> gives
$$
\boxed{\mathbb P(\text{a match with the designated member})=1-\left(1-\frac1m\right)^{n-1}.}
$$
For $n=23$, this is approximately $5.86\%$, while the probability of some matching pair is approximately $50.73\%$. The designated-member question examines $22$ pairs; the any-pair question examines all $253$. A repeated label elsewhere in the group does not require a match with its designated member.
\Image[/birthday-comparison-pairs-and-designated-member.png]
{title=Fifteen possible pairs versus five comparisons with one designated member}
{description=Original illustrative labels for six members: $(1,2,1,4,4,5)$. The <complete graph> includes every possible pair and marks the two equal-label pairs in red. The right panel uses only pairs containing member F, whose label 5 is unique. This chosen configuration illustrates the two different events; it is not a random sample of measured birthdays.}
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