Calculate the probability that at least two people in a group share the same birthday. Explore how counterintuitive probability can be.
Last updated: March 2026
| Group Size | Probability | Pairs |
|---|---|---|
| 5 | 2.71% | 10 pairs |
| 10 | 11.7% | 45 pairs |
| 23 | 50.7% ← Classic result | 253 pairs |
| 30 | 70.6% | 435 pairs |
| 50 | 97.0% | 1,225 pairs |
| 70 | 99.9% | 2,415 pairs |
The Birthday Paradox is the counterintuitive observation that in a relatively small group of randomly selected people, there's a surprisingly high probability that two will share the same birthday. It's called a "paradox" because the result contradicts most people's intuition about probability.
The key insight is that we shouldn't think about matching a specific person's birthday. Instead, we count all possible pairs within the group. With just 23 people, there are 253 unique pairs—each has a 1/365 chance of matching. This combinatorial explosion is why the probability reaches 50% with so few people.
The Classic Result: In a standard 365-day year with 23 randomly selected people, the probability that at least two share a birthday is approximately 50.7%. With 70 people, the probability exceeds 99.9%.
Input the number of people in your group. Start with 23 to see the classic example.
Change to 366 for leap years, or use different values to model other scenarios (e.g., 12 for months, 7 for days of the week).
The calculator displays the exact probability and key milestones (50%, 90%, 99%).
Why our intuition fails with 23 people:
It's paradoxical because the probability is counterintuitively high. Most people expect to need far more people to reach 50% probability, making the actual result seem surprising or contradictory.
Yes, the calculator assumes uniform distribution. In reality, births cluster around certain seasons and avoid February 29. These variations don't significantly change the core result.
With 366 days, you'd need about 24 people instead of 23 for 50% probability. The difference is minimal because the formula scales roughly as √(n).
Absolutely! Change 'Days' to model: 12 months (probability two people share a birth month), 7 days (share a day of the week), or even hash collisions in programming.
Demonstrates why collisions are inevitable in hashing, explains seeming coincidences in daily life, reveals cognitive biases about probability, and has applications in birthday attacks in cryptography.
Person 1 can pair with 22 others, Person 2 with 21, etc. Total pairs = 22+21+...+1 = 253. Formula: n(n-1)/2 = C(n,2).
If some people share genetic/geographic factors (families, regions), correlation increases probability. The calculator assumes independence, so actual results in families would be higher.
With enough pairs (pigeons) and limited possible values (holes), a collision is guaranteed. Here, 253 pairs competing for 365 days makes a match very likely.
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