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单词 BirthdayProblem
释义

birthday problem


The birthday problemMathworldPlanetmath is:What is the probability of two or more people out of a of n do have the same birthday?

Solution.Let A be the event two or more people out of a group of n to have the same birthday.It is easy to find first the complement of A, Ac, which is that no two people out of a group of n will have matching birthdays out of 365 (number of days of one year) equally possible birthdays. We start with an arbitrary person’s birthday. So the second person’s birthday in order to be different from the first one gives us the probability, 1-1365.Consecutively, the third person’s birthday is different from the first two and we have (1-1365)(1-2365). Finally for the the nth person’s birthday we get the probability

P[Ac]=(1-1365)(1-2365)(1-n-1365)
=(365-1)365(365-2)365(365-(n-1))365
=(365-1)365(365-(n-1))365n
=365!(365-n)!365n
=(365n)n!365n

By the last expression of the estimated probability we can see that there exists another easier approach to find the probability which is the following:

  1. 1.

    We set the possible ways for 1n birthdays, 365n as the denominator.

  2. 2.

    Using the binomial coefficientDlmfDlmfMathworldPlanetmath (365n), which represents the number of ways of picking n unordered birthdays from 365 days multiplied by the factorialMathworldPlanetmath n! used for the number of orders, in that way we determine the numerator.

Therefore we have

P[Ac]=(365n)n!365n

whence

P[A]=1-P[Ac]=1-(365n)n!365n.

The table below shows the probability P[A] for some values of n

nP[A]
100,11695
200,41144
230,50730
300,70632
400,89123
500,97037
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更新时间:2025/5/4 15:12:19