Birthday paradox
The statement saying that if a group of 23 or more people is given, then the probability that at least two of them have birthdays (day and month) will coincide, exceeds 50% . For a group of 60 or more people, the probability of coincidence of birthdays for at least two of its members is more than 99% , although it reaches 100% only when there are at least 366 people in the group (taking into account leap years - 367 ).
Such a statement may seem counterintuitive to common sense, since the probability of one being born on a certain day of the year is rather small, and the likelihood that two were born on a particular day is even smaller, but is true according to probability theory. Thus, it is not a paradox in a strict scientific sense - there is no logical contradiction in it, and the paradox consists only in the differences between the intuitive perception of a situation by a person and the results of a mathematical calculation.
Probability calculation
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Then the probability that at least two people from n birthdays coincide is equal to
The value of this function exceeds 1/2 with n = 23 (and the probability of coincidence is approximately 50.7%). The probabilities for some values of n are illustrated by the following table:
Close birthdays
Another generalization of the birthdays paradox consists in setting the problem of how many people are needed to ensure that the probability of having a group of people whose birthdays differ by no more than one day (or two, three days, etc.) exceeds 50%. . This task is more complicated; when it is solved, the inclusion-exclusion principle is used. The result (again, assuming that birthdays are evenly distributed) is obtained as follows:
Thus, the probability that even in a group of 7 people birthdays of at least two will differ by no more than a week, exceeds 50%.
Such a statement may seem counterintuitive to common sense, since the probability of one being born on a certain day of the year is rather small, and the likelihood that two were born on a particular day is even smaller, but is true according to probability theory. Thus, it is not a paradox in a strict scientific sense - there is no logical contradiction in it, and the paradox consists only in the differences between the intuitive perception of a situation by a person and the results of a mathematical calculation.
Probability calculation
')
Then the probability that at least two people from n birthdays coincide is equal to
The value of this function exceeds 1/2 with n = 23 (and the probability of coincidence is approximately 50.7%). The probabilities for some values of n are illustrated by the following table:
- n - p (n)
- 10 - 12%
- 20 - 41%
- 30 - 70%
- 50 - 97%
- 100 - 99.99996%
- 200 - 99.9999999999999999999999999998%
- 366 - 100%
Close birthdays
Another generalization of the birthdays paradox consists in setting the problem of how many people are needed to ensure that the probability of having a group of people whose birthdays differ by no more than one day (or two, three days, etc.) exceeds 50%. . This task is more complicated; when it is solved, the inclusion-exclusion principle is used. The result (again, assuming that birthdays are evenly distributed) is obtained as follows:
- The maximum difference of birthdays, days The required number of people
- 1 23
- 2 14
- 3 11
- 4 9
- 5 8
- 8 7
Thus, the probability that even in a group of 7 people birthdays of at least two will differ by no more than a week, exceeds 50%.
Source: https://habr.com/ru/post/72301/
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