Birthday Paradox Calculator

The 'birthday paradox' is the surprising fact that it only takes 23 people for there to be better-than-even odds that two of them share a birthday, even though there are 365 days to spread across. It's easiest to compute the complement first: the probability that everyone lands in a different category is (1) × (1 − 1/d) × (1 − 2/d) × ... × (1 − (n−1)/d), for a group of n people and d equally likely categories — the first person's category is free, and each person after that must avoid every category already claimed. The probability of at least one shared category is just 1 minus that product.

That product is computed in log-space (summing log(1 − i/d) terms and exponentiating at the end) so it stays numerically accurate for a large group instead of multiplying many factors that are each individually close to 1.

The 'number of possible categories' field defaults to 365 for the classic birthday version, but the underlying math applies to any set of equally likely categories — the same calculation answers 'what's the chance two people in a group share a favorite of N options' or 'how many random items until two probably collide into the same one of N buckets.' Once the group size exceeds the category count, a shared category is guaranteed outright by the pigeonhole principle, reported directly rather than computed from the product.