With 23 people in a room, a shared birthday is about as likely as not. The surprising part is how many pairs are quietly being compared.
We tend to compare everyone with ourselves, giving 22 chances for a match. But any two people can share a date. A group of 23 contains 253 distinct pairs, so there are far more opportunities for coincidence than intuition first notices.
Assume birthdays are spread evenly across 365 days and ignore leap day. It is easier to calculate the chance that nobody matches: each new person must avoid all birthdays already seen. Multiplying those successive chances gives a no-match probability of about 49.3%, so the chance of at least one shared birthday is about 50.7%. The 253 pair comparisons are not independent, so their probabilities cannot simply be multiplied.