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On the generalised birthday problem

  • R. J. McGregor (a1) and G. P. Shannon (a2)

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Probability theory abounds in counterintuitive results, perhaps the most celebrated being the answer to the birthday problem: what is the least value of n such that p (n, 2) > ½ where p (n, 2) denotes the probability that at least two out of n randomly chosen people have the same birthday? The question assumes birthdays are uniformly and independently distributed with leap years being ignored. The solution, n = 23, never fails to startle beginning students, and very often triggers an interest in the subject of probability. It is derived from the well-known observation that by the principle of complementation p (n, 2) is one minus the probability that no two have the same birthday, i.e.

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1. Feller, William, An introduction to probability theory and its applications, Wiley (1968).
2. Gehan, Edmund A., Note on the ‘Birthday Problem’, The American Statistician (1968) p. 28.

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