Birthday paradox
The assertion that if a group of 23 or more people is given, then the probability that at least two of them have the same birthdays (day and month) exceeds 50% . For a group of 60 or more people, the probability of coincidence of the birthday of at least two of its members is more than 99% , although it reaches 100% only when the group has at least 366 people (taking into account leap years - 367 ).
Such a statement may seem contrary to common sense, since the probability of one being born on a certain day of the year is rather small, and the likelihood of two being born on a particular day is even less, but is true in accordance with probability theory. Thus, it is not a paradox in the strict scientific sense - there is no logical contradiction in it, and the paradox lies only in the differences between a person’s intuitive perception of the situation and the results of mathematical calculation.
Calculation of probability.
Then the probability that at least two people out of n have the same birthday is equal.
The value of this function exceeds 1/2 for n = 23 (the probability of coincidence is approximately 50.7%). The probabilities for some values of n are illustrated in the following table:
Close birthdays
Another generalization of the birthday paradox is to pose the problem of how many people are needed to make it likely that there are people in the group whose birthdays differ by no more than one day (or two, three days, and so on), exceeded 50%. This task is more complicated, in its solution the principle of inclusion-exclusion 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 the birthdays of at least two will differ by no more than a week exceeds 50%.
Such a statement may seem contrary to common sense, since the probability of one being born on a certain day of the year is rather small, and the likelihood of two being born on a particular day is even less, but is true in accordance with probability theory. Thus, it is not a paradox in the strict scientific sense - there is no logical contradiction in it, and the paradox lies only in the differences between a person’s intuitive perception of the situation and the results of mathematical calculation.
Calculation of probability.
Then the probability that at least two people out of n have the same birthday is equal.
The value of this function exceeds 1/2 for n = 23 (the probability of coincidence is approximately 50.7%). The probabilities for some values of n are illustrated in the following table:
- n - p (n)
- 10 - 12%
- 20 - 41%
- 30 - 70%
- 50 - 97%
- 100 - 99.99996%
- 200 - 99.999999999999999999999999999998%
- 366 - 100%
Close birthdays
Another generalization of the birthday paradox is to pose the problem of how many people are needed to make it likely that there are people in the group whose birthdays differ by no more than one day (or two, three days, and so on), exceeded 50%. This task is more complicated, in its solution the principle of inclusion-exclusion is used. The result (again assuming that birthdays are evenly distributed) is obtained as follows:
- Maximum difference of birthdays, days 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 the birthdays of at least two will differ by no more than a week exceeds 50%.