The birthday paradox: what it is, and how to explain it
Let's imagine that we are with a group of people, for example, at a family reunion, a primary class reunion, or simply having a drink in a bar. Let's say there are about 25 people.
Between the noise and the superficial conversations, we have disconnected a bit 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 birthdays on the same day?
The birthday paradox is a mathematical truth, contrary to our instinct, which holds that very few people are needed for there to be a near random probability that two of them have the same birthday. Let's try to understand this curious paradox more thoroughly.
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The birthday paradox
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 the same birthday
. The popularity of this mathematical statement is due to the surprising fact that so few are necessary. people to have a pretty sure chance that they will have matches on something as varied as a birthday.Although this mathematical fact is called a paradox, in a strict sense it is not. It is rather a paradox insofar as it turns out to be curious, since it is quite contrary to common sense. When someone is asked how many people they think it takes for the two of them to have birthdays on the same day, people tend to intuitively give 183, that is, half of 365.
The thought behind this value is that by halving the number of days in an ordinary year, the minimum necessary for there to be a probability close to 50% is obtained.
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 probabilities that a specific person has a birthday with respect to another in the group, but, as we have commented, the chances that any two people in the group have the same birthday day.
Mathematical explanation of the phenomenon
To understand this surprising mathematical truth, the first thing to do is keep in mind that there are many possibilities to find couples who have the same birthday.
At first glance, one would think that 23 days, that is, the 23rd birthday of the band 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, as if to expect repetitions. This thinking is indeed accurate, but only if we expect a repeat on a particular day. That is to say, and as we have already commented, we would need to gather a lot of people so that there would be one more possibility or less close to 50% of one of the members of the group having a birthday with ourselves, to put a example.
However, in the birthday paradox any repetitions arise. That is, how many people are needed for two of those people to have their birthday on the same day, being the person or days any. To understand it and show it mathematically, Next we will see in more depth the procedure behind the paradox.
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Possibility of possible match
Let's imagine that we have only two people in a room. These two people, C1 and C2, could only form a couple (C1=C2), with which we only have one couple in which a repeat birthday can occur. Either they have their birthdays on the same day, or they don't have the same birthday, 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 a possible match
What happens if instead of two people there are three? Match chances 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 chances of a possible match
With four there are six possibilities that they coincide between them:
(4 people X 3 possible combinations)/2 = 6 chances of a possible match
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 one of them a candidate for the two members of it to have their birthday on the same day, giving themselves the birthday paradox and having more possibilities that there is 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, 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 probability opposite to what is stated in the birthday paradox. For this, We must take into account two possible events when considering the calculations.
Event A = {two people celebrate their birthdays on the same day} Complementary to the 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 follows the following calculation: 365 × 364 × 363 × 362 × 361 = 6,303 × 10^12, which gives as result is the number of cases where there are not two people in that group of 5 who were born 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 of two people in the group of 5 not having birthdays on the same day are 97.3%. With this data, we can obtain the possibility of two people having their birthday on the same day, obtaining the complementary value.
p(A) = 1 - p(A^c) = 1 - 0.973 = 0.027
Thus, from this it is extracted that the chances that in a group of five people, two of them have a birthday on the same day is only 2.7%.
Understanding this, we can change the size of the sample. The probability that at least two people in a gathering of n people have the same birthday can be obtained using the following formula:
1- ((365x364x363x…(365-n+1))/365^n)
In case n is 23, the probability that at least two of those people celebrate years on the same day is 0.51.
The reason why this specific sample size has become so famous is because with n = 23 there is an even probability that at least two people celebrate the 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 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 of understanding the problem is to pose it as follows.
Let's imagine that 23 people are together in a room and we want to calculate the chances that they do not share birthdays.
Suppose there is only one person in the room. The chances that everyone in the room will have different birthdays are obviously 100%, that is, probability 1. Basically, that person is alone, and since no one else is there, her birthday doesn't coincide with anyone else's.
Now another person walks in, and therefore there are two people in the room. The odds of her having a different birthday than the first person are 364/365, this is 0.9973 or 99.73%.
Enter a third. The probability that she has a different birthday than the other two people, who have entered before her, is 363/365. The odds that all three have different birthdays is 364/365 times 363/365, or 0.9918.
So, the options for 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% probability that none of those present have a birthday on the same day and, therefore, vice versa, calculating the complementary of that percentage we have that there is a 50.7% chance that at least two of them share birthday
In contrast to the birthday paradox, the probability that anyone in a room of n person birthday the same day as a specific person, for example, ourselves in case 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 Felipe VI of Spain, we have 20 legitimate monarchs. Among these kings we find, surprisingly, two couples who coincide on birthdays: Carlos II with Carlos 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 of that year, held in Tel Aviv, Israel, 26 countries participated, 24 of which They sent either solo singers or groups where the figure of the singer took on a special role. Among them, two singers coincided on a birthday: the representative of Israel, Kobi Marimi and the one from Switzerland, Luca Hänni, both celebrating their birthdays on October 8.
Bibliographic references:
- Abramson, M.; Moser, W. EITHER. 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