Birthday Paradox Calculator
Find the probability that at least two people in a group share a birthday — the famous, counterintuitive birthday problem.
Result
How to use this calculator
- Enter the number of people in the group.
- Leave days at 365 (or set 366 to include leap day).
- Read the probability that at least two people share a birthday.
- Try 23 for the classic ~50% result, or 70 for near-certainty.
About this calculator
The birthday paradox is the surprising fact that in a group of just 23 people, there is about a 50% chance that two of them share a birthday — and with 70 people it rises above 99.9%. It feels wrong because we instinctively think about how many people share our specific birthday, but the question is whether any two people match, and the number of possible pairs grows quadratically with group size. The calculation works by finding the probability that everyone's birthday is different — multiplying 365/365 × 364/365 × 363/365 and so on — and subtracting from one. This calculator computes that probability for any group size, assuming birthdays are spread uniformly across the year.
How it works — the formula
P(no match) = ∏_{i=0}^{k−1} (d − i) / d
P(at least one match) = 1 − P(no match)The chance every birthday is distinct is the product of each successive person avoiding all earlier birthdays; one minus that is the chance of at least one collision.
Worked examples
- Inputs:
- people=23, days=365
- Output:
- 50.73% chance of a shared birthday
- Inputs:
- people=50, days=365
- Output:
- 97.04%
- Inputs:
- people=70, days=365
- Output:
- 99.92%
Limitations
- Assumes uniform, independent birthdays across the year.
- Ignores leap days and real seasonal birth patterns.
- Limited to a single shared-birthday question, not triples or specific dates.
Idealized model; real-world probabilities are marginally higher due to birth clustering.
Frequently asked
Why is it called a paradox?+
How many people for a 50% chance?+
Why is the probability so high with so few people?+
How is the probability calculated?+
Does this account for leap years or birth seasonality?+
What happens with more than 365 people?+
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