Birthday Problem: Puzzle and Step-by-Step Solution

Step 1: Compute the complement

P(at least one shared birthday) is messy to compute directly, because the matches can overlap in many ways. P(all 23 birthdays are different) is easy. Ignore leap years and assume every day is equally likely.

The first person can have any birthday. The second must avoid one day, probability 364/365. The third must avoid two days, 363/365, and so on until the 23rd person avoids 22 days, 343/365. Multiplying the chain gives about 0.493. So the probability of at least one match is 1 - 0.493 = 0.507.

Step 2: A quick estimate you can do in your head

Each factor is 1 - k/365, and for small x, 1 - x is close to e^(-x). So the product is close to e^(-(1 + 2 + ... + 22)/365) = e^(-253/365) = e^(-0.693). And e^(-0.693) is 1/2, because ln 2 is 0.693. That is why 23 is the magic number.

The same approximation gives a general rule: with n people the chance of no match is about e^(-n²/730), so the 50% point sits near the square root of 2 × 365 × ln 2, which is about 22.5.

Step 3: Why intuition misses by so much

The 6%-of-the-calendar instinct compares 23 people with 365 days. But matches happen between pairs, and 23 people contain C(23, 2) = 253 pairs, each with a 1-in-365 chance of matching. Two hundred and fifty-three nearly independent chances at 1/365 land close to a half.

The count that matters grows with the square of the number of people, not linearly. The same effect is why hash collisions, duplicate trade IDs and shared PINs appear far sooner than people expect.

A quick check on the size: with 57 people the probability of a shared birthday is above 99%, and with 70 people it is about 99.9%. Being able to quote those numbers, or estimate them quickly from e^(-n²/730) without a calculator, is a common follow-up in trading interviews.