Birthday Paradox Calculator

The birthday paradox asks how likely it is that at least two people in a group share the same birthday. The result feels surprising because people often compare one chosen person with everyone else. The actual question compares every possible pair. A group of 23 contains 253 pairs, so there are many opportunities for a match even though the calendar has 365 possible birthdays. Under the standard model, a group of 23 already has a probability slightly above 50 percent of containing a shared birthday. The cleanest calculation starts with the complementary event: every person has a different birthday. The first person may have any birthday. The second must avoid that date, giving a factor of 364/365. The third must avoid two occupied dates, giving 363/365, and the multiplication continues for the whole group. Subtracting that product from one gives the probability of at least one match. In symbols, P(match) = 1 - (365/365)(364/365)...((365-n+1)/365), where n is the group size. This calculator uses the conventional simplified birthday model. It treats all 365 calendar dates as equally likely, ignores leap day, and assumes births are independent. Real birth records are not perfectly uniform because seasonality, holidays, scheduled deliveries, and demographic patterns affect daily birth rates. Those differences generally make collisions a little more likely, but the standard model is the appropriate version for probability lessons, interview questions, classroom demonstrations, and rough planning. The probability rises quickly. With 10 people it is about 11.7 percent, with 23 it is about 50.7 percent, with 30 it is about 70.6 percent, and with 50 it exceeds 97 percent. A group needs 57 people to pass 99 percent. This rapid increase illustrates a broader collision principle used in hashing, cryptography, database identifiers, and quality control: when many items are assigned to a limited number of categories, duplicates become likely much sooner than intuition suggests. Use the result as a probability, not a prediction that identifies who will match or which date will repeat. A 50.7 percent probability means that across many comparable groups of 23, just over half would contain at least one shared birthday. One particular group may have no match, one matching pair, or several repeated dates. The calculator answers the at-least-one-match question exactly for the assumptions above.