Birthday Probabilities

The birthday paradox is a mathematical problem put forward by Von Mises. It answers the question: what is the minimum number $ N $ of people in a group so that there is a 50% chance that at least 2 people share the same birthday (day-month couple). The answer is $ N = 23 $, which is quite counter-intuitive, most people estimate this number to be much larger, hence the paradox.

In the following FAQ, a year has 365 days (calendar leap years are ignored).

Probability is $ 1/365 \approx 0.0027 \approx 0.27\% $, indeed, 1 chance out of 365 to be born on a precise day in a year with 365 days.

NB: the complementary (opposite) probability of not being born on a certain day of the year is $ 1-1/365 = 364/365 \approx 0.9973 \approx 99.73\% $, indeed, there are 364 possible days out of 365.

By taking 2 people at random, and noting them A and B, this calculation amounts to asking the question What is the probability that B was born on a certain day of the year? this certain day being the birthday of birth of A. The probability is always $ 1/365 \approx 0.0027 \approx 0.27\% $

Calculating this probability is equivalent to calculating the opposite of the probability that all people were born on a different day (because in this case at least 2 would be born on the same day).

For the smallest of the groups: N = 2. This calculation is explained above in What is the probability that 2 people were born on the same day? so $ P(N=2) \approx 0.27\% $. Note that this probability is the opposite of the probability that a person A was born on a different day from a person B and can therefore also be calculated $$ P(N=2) = 1 - (364/365) = 0.0027 = 0.27\% $$

For a larger group, N=3 composed of people A, B and C. There are therefore 364 chances out of 365 that B was not born on the same day as A and 363 chances out of 365 that C was not born the same days as A and B. Thus, $$ P(N=3) = 1 - (364/365) \times (363/365) \approx 0.82\% $$

In the same way, $$ P(N=4) = 1 - (364/365) \times (363/365) \times (362/365) \approx 1.64\% \\ P(N=5) = 1 - (364/365) \times \cdots \times (361/365) \approx 2.71\% $$

The general formula is $$ P(N=n) = 1 - \left(\frac{364}{365}\right) \times \left(\frac{363}{365}\right) \times \cdots \times \left(\frac{365-(n-1)}{365}\right) $$

A group of 23 people has a probability of approximately 0.5 or 50% (1 chance out of 2) that two people have a common birth date (day + month).

$$ P(N=23) = 1 - \left(\frac{364}{365}\right) \times \left(\frac{363}{365}\right) \times \cdots \times \left(\frac{343}{365}\right) \approx 0.5073 $$

The probability that no one shares a birthday is the opposite of the probability that there are (at least) 2 in common

$$ P(N=n) = \left(\frac{364}{365}\right) \times \left(\frac{363}{365}\right) \times \cdots \times \left(\frac{365-(n-1)}{365}\right) $$

By taking N people at random numbered from 1 to N, then by denoting D the date of birth of the first person, this amounts to calculating the probability that N-1 other people were born on date D. For person 2, the probability is $ P = 1/365 \approx 0.0027 \approx 0.27\% $, for person 3, same proba, $ P = 1/365 $, etc. The probabilities are multiplied, i.e. the formula:

$$ P(X=N) = \left( \frac{1}{365} \right)^{N-1} $$

Calculate the opposite of the probability that the N people were born on another day.

$$ P(N=n) = 1 - (364/365)^{n-1} $$

For a probability of 50%, a minimum of 253 people will be needed that the probability that 2 were born on a specific day is approximately 1/2.

NB: The complementary probability that all N people are not born on a given date is P = (364/365)^N

366 are needed, ie, one for every day of the year plus one to be sure that at least a couple of two people share the same birthday.

On average, about $ \frac{1}{365} $ of the world's population shares the same birthday.

With the hypothesis of a world population of 8 billion people. There are 8000000000/365 or about 22 million people born on a given day+month.

Limiting to the USA (350 millions inhabitants), there are 350000000/365 or nearly 1 millions US people born on a given day+month date.