Probability of Matching Birthdays

(1) Probability that none of the birthdays in the group match

\(\normalsize \\ \hspace{20px}\overline{p}(n)={\large\frac{364}{365}}\times{\large\frac{363}{365}}\times{\large\frac{362}{365}}\times\cdots\\ \hspace{50px}={\large\frac{365\times364\times363\times\cdots\times(365-n+1)}{365^n}}\\ \hspace{50px}={\large\frac{365!}{365^n(365-n)!}}\\\)

(2) Probability that at least one pair of birthdays matches within a group

\(\normalsize\\ \hspace{20px}p(n)=1-\overline{p}(n)\\\)

The reason the probability of a match appears high is that people tend to think in terms of the probability that their own birthday will match.
This phenomenon is known as the “Birthday Paradox.” Please note that leap years are not taken into account.