The Birthday Paradox: what it is, and how it is explained

Let's see what this curious mathematical phenomenon consists of, and what it is due to.

Let's imagine that we are with a group of people, for example, at a family reunion, a reunion of the elementary school class, or simply having a drink in a bar. Let's say there are about 25 people.

Amidst the hubbub and superficial conversations, we've tuned out a bit and started minding our own business, and suddenly we ask ourselves: what must be the probability that among these people, two people have the same birthday?

The birthday paradox is a mathematical truth, contrary to our instinct, contrary to our instinct, contrary to our instinct.The birthday paradox is a mathematical truth, contrary to our instinct, which holds that it takes very few people to have a close-to-random probability that two of them have the same birthday. Let's try to understand this curious paradox further.

The birthday paradox

The birthday paradox is a mathematical truth that states that in a group of just 23 people there is a probability close to random, namely 50.7%, that at least two of those people will have the same birthday on the same day.. The popularity of this mathematical statement is due to the surprising fact that it takes so few people to have a fairly certain chance of having coincidences in something as varied as birthdays.

Although this mathematical fact is called a paradox, strictly speaking it is not. It is rather a paradox in that it happens to be a curious oneIt is rather a paradox in that it is quite contrary to common sense. When you ask someone how many people they think it takes for two people to have the same birthday on the same day, people tend to intuitively answer 183, that is, half of 365.

The thinking behind this value is that halving the number of days in an ordinary year provides the minimum necessary for a probability of close to 50%.

However, it is not surprising that such a high value is given when trying to answer this questionThe birthday paradox does not refer to the birthday paradox, since people tend to misunderstand the problem. The birthday paradox does not refer to the probabilities of a particular person having a birthday relative to another person in the group, but, as we have discussed, the chances of any two people in the group having the same birthday on the same day.

Mathematical explanation of the phenomenon

To understand this surprising mathematical truth, the first thing to keep in mind is that there is a good chance of finding couples with the same birthday.

At first glance, one would think that 23 days, that is, the 23 birthdays of the group members, is too small a fraction of the possible number of different days in a non-biparous year.365 days in a non-leap year, or 366 in leap years, to expect repetitions. This thought is actually accurate, but only if we would expect the repetition of a particular day. That is, as we have already mentioned, we would need to gather a lot of people for there to be a more or less 50% chance that one of the members of the group would have a birthday with ourselves, for example.

However, in the birthday paradox any repetition arises. That is, how many people does it take for two of those people to have the same birthday on the same day, being any one person or any two days. To understand it and to show it in a mathematical way, we will now take a closer look at the procedure behind the paradox..

Possibilities of possible coincidence

Let's imagine that we have in a room only two people. These two people, C1 and C2, could only form a couple (C1=C2), so we have only one couple in which there could be a repetition of birthdays. Either they have the same birthday, or they do not have the same birthday, there are no other alternatives..

To state this fact mathematically, we have the following formula:

(Nº people x possible combinations)/2 = possibilities of possible coincidence.

In this case, this would be:

(2 x 1)/2 = 1 possibility of possible match.

What if instead of two people there are three? The possibilities of coincidence increase to threeThe chances of coincidence increase to three, thanks to the fact that three pairs can be formed between these three persons (Cl=C2; Cl=C3; C2=C3). Mathematically represented we have:

(3 persons X 2 possible combinations)/2 = 3 possibilities of possible coincidence.

With four there are six possibilities of matching between them:

(4 people X 3 possible combinations)/2 = 6 possibilities of possible coincidence.

If we go up to ten people, we have many more possibilities:

(10 people X 9 possible combinations)/2 = 45

With 23 people there are (23×22)/2 = 253 different coupleseach one of them a candidate to have both members of the couple with the same birthday, giving the birthday paradox and more possibilities for the birthday coincidence.

Probability estimation

Let's calculate what is the probability that a group with size n of people two of them have the same birthday.all birthdays, whatever they are, have the same birthday. For this particular case, we will discard leap years and twins, assuming that there are 365 birthdays that have the same probability.

Using Laplace's rule and combinatorics

First, we have to calculate the probability that n people have different birthdays. That is, we calculate the opposite probability of what is posed by the birthday paradox. For this we must take into account two possible events, we have to take into account two possible events when performing the calculations.

Event A = {two people celebrate their birthday on the same day}. Complementary to event A: A^c = {two people do not celebrate their birthday on the same day}.

Let's take as a particular case a group with five people (n=5).

To calculate the number of possible cases, we use the following formula:

Days of the year^n

Taking into account that a normal year has 365 days, the number of possible cases of birthday celebration is:

365^5 = 6,478 × 10^12

The first of the people we select may have been born, as is logical to think, on any of the 365 days of the year. The next one may have been born on one of the remaining 364 days of the year.The next of the next may have been born on one of the remaining 363 days, and so on.

From this follows the following calculation: 365 × 364 × 364 × 363 × 362 × 361 = 6.303 × 10^12, which gives the number of cases where no two people in that group of 5 were born on the same day.

Applying Laplace's rule, we would calculate:

P (A^c) = favorable cases/possible cases = 6.303 / 6.478 = 0.973.

This means that the chances that two people in the group of 5 do not have the same birthday are 97.3%.. With this data, we can obtain what is the possibility that two people have the same birthday, obtaining the complementary value.

p(A) = 1 - p(A^c) = 1 - 0.973 = 0.027

Thus, it follows that the chances that in a group of five people, two of them will have the same birthday is only 2.7%.

Having understood this, we can change the sample size. The probability that at least two people in a gathering of n people will have the same birthday on the same day can be obtained by the following formula:

1- ((365x364x363x…(365-n+1))/365^n)

In case n is 23, the probability that at least two of those people will have the same birthday is 0.51.

The reason why this particular sample size has become so famous is because at n = 23 there is an even probability that at least two people will celebrate the same birthday on the same day.

If we increase to other values, for example 30 or 50, we have higher probabilities, of 0.71 and 0.97 respectively, or what is the same, 71% and 97%. With n = 70 we are almost assured that two of them will coincide on their birthday, with a probability of 0.99916 or 99.9%.

Using Laplace's rule and the product rule

Another not so far-fetched way to understand the problem is to pose it as follows..

Let's imagine that 23 people are gathered in a room and we want to calculate the chances that they do not share a birthday.

Suppose there is only one person in the room. The chances that everyone in the room has a birthday on different days are obviously 100%, i.e. probability 1. Basically, that person is alone, and since there is no one else his birthday does not coincide with anyone else's birthday.

Now another person enters and, therefore, there are two people in the room. The odds of having a birthday different from the first person's are 364/365that is 0.9973 or 99.73%.

Enter a third person. The probability that she has a different birthday from the other two people, who have entered before her, is 363/365. The odds of all three having different birthdays is 364/365 times 363/365, or 0.9918.

Thus, the odds of 23 people having different birthdays are 364/365 x 363/365 x 362/365 x 361/365 x 361/365 x ... x 343/365, resulting in 0.493.

That is, there is a 49.3% chance that none of those present will have the same birthday and, therefore, conversely, calculating the complementary of that percentage we have a 50.7% chance that at least two of them will share the same birthday.

In contrast to the birthday paradox, the probability that anyone in a room of n people has a birthday on the same day as a particular person, e.g., ourselves in case we are there, is given by the following formula.

1- (364/365)^n

With n = 23 it would give about 0.061 probability (6%), needing at least n = 253 to give a value close to 0.5 or 50%.

The paradox in reality

There are many situations in which we can see that this paradox is fulfilled. Here we are going to put two real cases.

The first is that of the kings of Spain.. Counting from the reign of the Catholic Kings of Castile and Aragon to that of Philip VI of Spain, we have 20 legitimate monarchs. Among these kings we find, surprisingly, two couples that coincide in birthdays: Charles II with Charles IV (November 11) and Joseph I with Juan Carlos I (January 5). The chance that there would be only one pair of monarchs with the same birthday, taking into account that n = 20, is

Another real case is that of the 2019 Eurovision grand final.. In the final of that year, held in Tel Aviv, Israel, 26 countries came to participate, 24 of which sent either solo singers or groups where the figure of the singer took special prominence. Among them, two singers coincided on a birthday: the representative of Israel, Kobi Marimi and that of Switzerland, Luca Hänni, both having their birthdays on October 8.

Bibliographical references:

• Abramson, M.; Moser, W. O. J. (1970). "More Birthday Surprises". American Mathematical Monthly. 77 (8): 856-858. doi:10.2307/2317022.
• Bloom, D. (1973). "A Birthday Problem". American Mathematical Monthly. 80 (10): 1141-1142. doi:10.2307/2318556.
• Klamkin, M.; Newman, D. (1967). "Extensions of the Birthday Surprise". Journal of Combinatorial Theory. 3 (3): 279-282. doi:10.1016/s0021-9800(67)80075-9