Matching one specified birthday ($1-(1-1/m{)}^{n-1}$)

In a group of total size $n\ge 1$, 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 givesFor $n=23$, 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.