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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.

A graph showing the probability of at least two people sharing a birthday amongst a certain number of people. The step-like appearance is due to the x axis taking only integer values, since it indicates numbers of people.

Contents

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• 1 Understanding the paradox
• 2 Calculating the probability
• 3 Approximations
  • 3.1 A simple exponentiation
  • 3.2 Poisson approximation
  • 3.3 Approximation of number of people
• 4 An upper bound and a different perspective
• 5 Generalization
  • 5.1 Applications
• 6 Other birthday problems
  • 6.1 Reverse problem
    • 6.1.1 Sample calculations
  • 6.2 Same birthday as you
  • 6.3 365 different birthdays
  • 6.4 Near matches
  • 6.5 Collision counting
• 7 References
• 8 Notes
• 9 External links

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