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Birthday Paradox Probability
23 people · 50.73% chance of shared birthday
Generated
7/8/2026
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%).
Quick reference
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).
Why it's called a paradox: intuition says you need ~183 people (half of 365) for a 50% chance — but the answer is 23. The trick is that with n people there are n×(n−1)/2 PAIRS, and each pair is a chance to match. With 23 people that's 253 pairs. Used in cryptography (birthday attack on hash functions).