Birthday Paradox Calculator
Compute the probability that 2 or more people in a group share a birthday. The famous mathematical 'paradox' explained with chart.
Fun
Birthday Paradox Calculator
Generated on April 25, 2026
Probability of shared birthday
50.73%
In a random group of 23 people, there's a 50.7% chance that at least two share a birthday.
Probability vs group size
Pink dot = your group (23 people). Amber dot = the famous threshold (23 people, 50.7%).
Step-by-step calculation
Formula
P(shared) = 1 − ∏ (365 − i) / 365 for i = 0 to n−1
- 1Group size: n = 23.
- 2Easier to compute the COMPLEMENT — probability that everyone has a unique birthday.
- 3Person 1 picks any of 365 days. Person 2 must avoid that one (364/365). Person 3 avoids both (363/365). And so on.
- 4P(all unique) = (365/365) × (364/365) × … × ((365−n+1)/365)
- 5P(at least one shared) = 1 − P(all unique) = 50.7297%
- 6The famous threshold: at n = 23, the probability crosses 50% (50.73% to be exact).
?What is the Birthday Paradox Calculator?
The Birthday Paradox Calculator computes the probability that at least two people in a group of N share a birthday — the famous result that surprises everyone with how few people are needed for a 50/50 chance: just 23. The 'paradox' isn't a logical contradiction; it's a misalignment between intuition and combinatorics. With 23 people there are 23×22÷2 = 253 unique pairs, and each pair has a 1/365 chance of matching — 253 chances for a match adds up faster than your gut suggests. Used in cryptography to estimate the difficulty of finding hash collisions (the 'birthday attack').
The Formula
Direct counting of shared birthdays is hard, so we compute the COMPLEMENT: probability that everyone has a unique birthday. Person 1 picks any day (365/365 = 1). Person 2 must avoid Person 1's day (364/365). Person 3 must avoid both (363/365). And so on. Multiplying these gives P(all unique); subtracting from 1 gives P(at least one match). The probability rises faster than intuition suggests because the number of pairs grows quadratically with group size — n(n-1)/2 pairs in a group of n.
Birthday Paradox — Probability of Shared Birthday by Group Size
Probability that at least two people in a group share a birthday.
| Group size | Probability | Comparison |
|---|---|---|
| 5 | 2.71% | Small dinner party |
| 10 | 11.69% | Small office team |
| 15 | 25.29% | Sports team starting lineup |
| 20 | 41.14% | Average book club |
| 23 | 50.73% | FAMOUS THRESHOLD — 50/50 |
| 30 | 70.63% | Typical school class |
| 40 | 89.12% | Wedding party |
| 50 | 97.04% | Conference workshop |
| 70 | 99.92% | Large birthday party |
| 100 | 99.9999% | Practically certain |
Practical Examples
5 people: 2.7% chance of a shared birthday.
10 people: 11.7% chance.
20 people: 41.1% chance.
23 people: 50.7% chance — the famous threshold.
30 people (typical school class): 70.6% chance — most classrooms have at least one shared birthday.
50 people: 97.0% chance.
70 people: 99.92% chance — practically guaranteed.
Cryptographic 'birthday attack' on a 256-bit hash needs only ~2^128 operations, not 2^256, to find a collision.
Frequently Asked Questions
Popular Conversions
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