Birthday paradox

### From Wikipedia, the free encyclopedia

In probability theory, the **birthday paradox** states that given a group of 23 (or more) randomly chosen people, the probability is more than 50% that some pair of them will have the same birthday. For 57 or more people, the probability is greater than 99%, although it cannot be exactly 100% unless there are at least 366 people.^{[1]} This is not a paradox in the sense of leading to a logical contradiction; it is called a paradox because mathematical truth contradicts naive intuition: most people estimate that the chance is much lower than 50%. Calculating this probability (and related ones) is the **birthday problem**. The mathematics behind it has been used to devise a well-known cryptographic attack named the birthday attack.

## Contents

[hide]- 1 Understanding the paradox
- 2 Calculating the probability
- 3 Approximations
- 4 An upper bound and a different perspective
- 5 Generalization
- 6 Other birthday problems
- 7 References
- 8 Notes
- 9 External links

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