Hong Life CalculatorsUnderstanding the birthday paradox calculator — concept and principle
A birthday paradox calculator finds the probability that, in a group of k people, at least two share the same birthday. The answer surprises most people: only 23 people are needed for a better-than-even chance. The same mathematics is used to estimate collision risk in hash functions, random ID generation and sampling with replacement.
How it is calculated
- Probability that all k birthdays differ: P(no match) = (d/d) × ((d − 1)/d) × ((d − 2)/d) × … × ((d − k + 1)/d), the product of (1 − i/d) for i = 0 to k − 1, where d is the number of days in the year (365 by default).
- P(at least two share) = 1 − P(no match), also shown as a percentage.
If k is larger than d, a match is certain (the pigeonhole principle), and the product becomes 0.
The principle
It is easier to count the opposite event. The first person can have any birthday; the second must avoid one day, the third must avoid two, and so on. Multiplying these shrinking chances gives the probability that everyone is different. The result feels paradoxical because people instinctively compare against their own birthday, but a match can happen between any pair. With 23 people there are 23 × 22 ÷ 2 = 253 pairs, and each pair has a 1/365 chance of matching. A handy approximation is P(match) ≈ 1 − e^(−k(k−1)/(2d)).
Worked example
- Inputs: k = 23 people, d = 365 days.
- P(no match) = 365/365 × 364/365 × … × 343/365 ≈ 0.49270.
- P(at least two share) = 1 − 0.49270 ≈ 0.5073, or 50.73%.
- Check with the approximation: 1 − e^(−253/365) ≈ 0.500, close to the exact value.
| People in the room | P(at least two share) |
|---|---|
| 5.75 | 0.0404624837 |
| 11.5 | 0.1670247888 |
| 17.2 | 0.3469114179 |
| 23 | 0.5072972343 |
| 28.8 | 0.6809685375 |
| 34.5 | 0.8143832389 |
| 46 | 0.9482528434 |
How to read the result
The probability rises fast: about 11.7% for 10 people, 70.6% for 30, 97.0% for 50 and 99.9% for 70. In contrast, the chance that someone in a group of 23 shares your own birthday is only about 6%. For computer systems, the same rule says that a hash with N possible values starts to see collisions after roughly √N items.
Common mistakes
- Confusing "any two people match" with "someone matches me".
- Assuming 183 people (half of 365) are needed for a 50% chance.
- Forgetting that the result is for at least one shared birthday, not exactly one pair.
Frequently asked questions
How many people are needed for a 99% chance?
57 people give about 99.0%.
Does including 29 February change the answer?
Only slightly. Set the days to 366 to see the effect; 23 people still give about 50%.
What is the birthday attack in cryptography?
It uses this same probability to find two inputs with the same hash much faster than trying to match one fixed hash.
Sources: Feller, W., An Introduction to Probability Theory and Its Applications, Vol. 1 (Wiley) · Ross, S. M., A First Course in Probability (Pearson)