HesapHanem

Birthday Paradox Calculator

See the shared-birthday probability.

How it is calculated

The tool finds the chance that at least two people in a group share a birthday. It first computes the chance that everyone's birthday is different: Q = (365/365) × (364/365) × … × ((365 − n + 1)/365), multiplying one factor per person, then reports P = 100 × (1 − Q) as a percentage with two decimals. The model assumes 365 equally likely birthdays, with no 29 February and no seasonal clustering, so real groups differ slightly. n is the number of people and must be a whole number from 2 to 365; 0, 1, fractions and anything above 365 give no result. At 365 people the display reads 100.00 % (the true value is just below 100 %, rounded).

Worked example

Take a group of 23 people. The first has any birthday (365/365 = 1). The second differs from the first with probability 364/365, the third 363/365, and so on down to 343/365 for the 23rd. Multiplying all 23 factors gives Q ≈ 0.4927, the chance that nobody shares. Then 1 − 0.4927 = 0.5073. The calculator shows 50.73%, so a group of just 23 people has slightly better than even odds of a shared birthday.

Try this example →

Common mistakes

  • Thinking you need about 183 people (half of 365) for a 50 % chance. The count of pairs grows much faster than the group, so 23 is enough.
  • Reading the result as the chance that someone shares your birthday. This is the chance that any two people match, a much easier event.
  • Counting only matches with one fixed person. The tool compares every pair in the group.
  • Expecting leap days or uneven birth rates to be included. The tool assumes 365 equally likely dates.
  • Entering a number of birthdays or years instead of the number of people.

About this calculator

The birthday paradox asks how likely it is that at least two people in a group share a birthday, and the answer is famously counterintuitive: with just 23 people, the probability already exceeds 50%, and with 70 people it’s over 99.9%, even though there are 365 possible birthdays. The exact formula computes the probability of no shared birthday first — 365 ÷ 365 × 364 ÷ 365 × 363 ÷ 365, and so on for each additional person — then subtracts that from 1, since it’s the number of pairs of people, which grows much faster than the group size, that drives the effect.

The same birthday-problem math underlies the birthday attack in cryptography, where finding any collision in a hash function turns out to require far fewer attempts than finding a specific match, which is why hash output lengths are chosen with this square-root-scaling effect in mind. It’s also a standard teaching example in probability courses for demonstrating how intuition about random matching can badly mislead you.

Was this helpful?

You can pick up to 3 reactions.

Comments

  • Be the first to comment.

Popular calculators

Loan Calculator

Monthly payment, total interest and full amortization schedule.

Percentage Calculator

Percent change, percent of, and ratio calculations.

Discount Calculator

Calculate final price, savings, and discount amount.

BMI Calculator

Body mass index from height and weight (metric or imperial).

Age Calculator

Compute your exact age in years, months and days from a birth date.

Compound Interest Calculator

Calculate the final amount and interest earned with compounding.

All Calculators