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birthday problem


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A well-known, but intriguing probability problem. There are n people in a room. Assume that none is born on 29 February and that the remaining 365 days are all equally likely as birthdays. What is the smallest value of n for which the probability that at least two have the same birthday is greater than 0.5?

The answer is not 183, but 23. The complementary event is that all n people have different birthdays. The probability of this, pn, is which reduces surprisingly quickly as n increases:

n

3

5

9

13

16

19

22

23

26

30

34

40

46

pn

0.99

0.97

0.91

0.81

0.72

0.62

0.52

0.49

0.40

0.29

0.20

0.11

0.05

Subjects: Probability and Statistics.


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