The Birthday Paradox: What It Is, And How It Is Explained
Let’s imagine that we are with a group of people, for example, at a family reunion, a primary school reunion, or simply having a drink at a bar. Let’s say there are about 25 people.
Between the noise and the superficial conversations, we have disconnected a little and we have started to think about our things and, suddenly, we ask ourselves: what must be the probability that among these people two people have a birthday on the same day?
The birthday paradox is a mathematical truth, contrary to our instinct, which maintains that there are very few people necessary for there to be a close to random probability of two of them having a birthday on the same day. Let’s try to understand this curious paradox more thoroughly.
The birthday paradox is a mathematical truth that establishes that in a group of just 23 people there is a probability close to chance, specifically 50.7%, that at least two of those people have a birthday on the same day The popularity of this mathematical statement is due to the surprising fact that so few people are needed to have a fairly safe chance of having matches on something as varied as the date of birth.
Although this mathematical fact is called a paradox, in a strict sense it is not. It is rather a paradox in that it turns out to be curious, since it is quite contrary to common sense. When you ask someone how many people they think it takes for two of them to have a birthday on the same day, people tend to give, as an intuitive answer, 183, that is, half of 365.
The thinking behind this value is that by halving the number of days that an ordinary year has, the minimum necessary is obtained so that there is a probability close to 50%.
However, It is not surprising that such high values are given when trying to answer this question, since people often misunderstand the problem. The birthday paradox does not refer to the chances of a specific person having a birthday compared to another person in the group, but, as we have mentioned, the chances of any two people in the group having a birthday on the same day.
Mathematical explanation of the phenomenon
To understand this surprising mathematical truth, what you must first do is keep in mind that there are many possibilities of finding couples who have a birthday on the same day.
At first glance, one would think that 23 days, that is, the 23rd birthdays of the group members, is too small a fraction of the possible number of distinct days, 365 days of a non-leap year, or 366 in leap years, to expect repetitions. This thought really is accurate, but only if we expected the repetition of a specific day. That is to say, and as we have already mentioned, we would need to bring together a lot of people so that there would be a more or less 50% chance that any of the members of the group would have a birthday with ourselves, to give an example.
However, in the birthday paradox any repetitions arise. That is, how many people are needed for two of those people to have a birthday on the same day, being the person or any given day. To understand it and show it mathematically, Next we will see in more depth the procedure behind the paradox
Possible match possibilities
Let’s imagine that we have only two people in a room. These two people, C1 and C2, could only form one couple (C1=C2), so we only have one couple in which a repeat birthday can occur. Either they have their birthday on the same day, or they don’t have their birthday on the same day, there are no other alternatives
To state this fact mathematically, we have the following formula:
(No. of people x possible combinations)/2 = possibilities of possible coincidence.
In this case, this would be:
(2 x 1)/2 = 1 chance of possible match
What happens if instead of two people there are three? The chances of matching go up to three, thanks to the fact that three pairs can be formed between these three people (Cl=C2; Cl=C3; C2=C3). Mathematically represented we have:
(3 people X 2 possible combinations)/2 = 3 possibilities of possible coincidence
With four there are six possibilities that they coincide:
(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 couples each of them a candidate for their two members’ birthdays on the same day, giving rise to the birthday paradox and there being more possibilities of a birthday coincidence.
Probability estimation
We are going to calculate what is the probability that a group with size n of people, two of them Whatever they are, they have their birthday on the same day. For this specific case, we are going to 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 to what is stated in the birthday paradox. For this, We must take into account two possible events when making the calculations
Event A = {two people celebrate their birthdays on the same day} Complementary to event A: A^c = {two people do not celebrate their birthdays 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 celebrations 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 in one of the remaining 364 days and the next of the next may have been born in one of the remaining 363 days, and so on.
From this the following calculation follows: 365 × 364 × 363 × 362 × 361 = 6.303 × 10^12, which results in the number of cases in which there are no two people in that group of 5 who were born on the same day.
This means that the chances that two people in the group of 5 will not have a birthday on the same day are 97.3% With this data, we can obtain the possibility of two people having a birthday on the same day, obtaining the complementary value.
p(A) = 1 – p(A^c) = 1 – 0.973 = 0.027
So, from this we can conclude that the chances that in a group of five people, two of them having a birthday on the same day is only 2.7%.
Once this is understood, we can change the sample size The probability that at least two people in a gathering of n people have a birthday on the same day can be obtained using the following formula:
1- ((365x364x363x…(365-n+1))/365^n)
If n is 23, the probability that at least two of those people celebrate birthdays on the same day is 0.51.
The reason why this particular sample size has become so famous is because with n = 23 There is an even chance that at least two people will celebrate their birthday on the same day
If we increase to other values, for example 30 or 50, we have higher probabilities, 0.71 and 0.97 respectively, or what is the same, 71% and 97%. With n = 70 we are almost guaranteed 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 in the following way
Let’s imagine that 23 people get together in a room and we want to calculate the chances of them not sharing a birthday.
Suppose there is only one person in the room. The chances of everyone in the room having a birthday on different days are obviously 100%, that is, probability 1. Basically, that person is alone, and since there is no one else, his or her birthday does not coincide with that of nobody else.
Now another person comes in and therefore there are two people in the room. The odds that you have a different birthday than the first person are 364/365 this is 0.9973 or 99.73%.
A third enters. The probability that she has a different birthday than the other two people, who 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 chances of 23 people having different birthdays are 364/365 x 363/365 x 362/365 x 361/365 x … x 343/365, resulting in 0.493.
In other words, there is a 49.3% chance that none of those present will have a birthday on the same day and, therefore, conversely, calculating the complement of that percentage, we have a 50.7% chance that at least two of them share birthdays.
In contrast to the birthday paradox, the probability that anyone in a room of n people will have the same birthday as a specific person, for example, ourselves if we are there, is given by the following formula
1- (364/365)^n
With n = 23 it would give around 0.061 probability (6%), requiring at least n = 253 to give a value close to 0.5 or 50%.
The paradox in reality
There are multiple 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 Monarchs of Castile and Aragon to that of Philip VI of Spain, we have 20 legitimate monarchs. Among these kings we find, surprisingly, two couples who coincide in birthdays: Charles II with Charles IV (November 11) and José I with Juan Carlos I (January 5). The possibility that there was 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 that year, held in Tel Aviv, Israel, 26 countries participated, 24 of which sent either solo singers or groups in which the figure of the singer took on special prominence. Among them, two singers coincided on their birthdays: the representative of Israel, Kobi Marimi, and that of Switzerland, Luca Hänni, both having their birthdays on October 8.
