As humans, despite numerous things to fascinate us, we always become shocked when we meet someone who shares the same birthday as us. While this level of surprise may come across as understandable if you are one of two people in a given setting, it wouldn’t be so if you were one student among 23 in a classroom, or one of 23 people in a cafe you’ve sat in to drink a hot chocolate or peach-mango tea, or one of the 23 people on the Argentinian national team for the 2022 World Cup.
That is because math tells us that if there are at least 23 people in a given setting, the likelihood of two of them sharing the same birthday is over 50%. This possibility is as likely as a tossed coin landing on heads. In fact, if there are 60 or more people in a given setting, there is a 99% chance that you will meet someone who shares the same birthday as you.
However, this situation continues to befuddle everyone who hears of it for the first time and invokes the question of “how is that possible?” because there are 365 days in the solar calendar. It seems more likely to us that everyone’s birthday falls on separate dates. Furthermore, the probability of sharing the same birthday with one of any 22 people who enter our lives in a given instance should be utterly low in a system where there are 365 days. For instance, while the probability of a tossed coin landing on heads or tails being only 50%, it would seem that the probability of two of 23 random selections out of a possible total of 365 should not exceed 50%.
Yet, normally the human brain has trouble with exponential thinking, and this paradox is approached intuitively. Because we are often mistaken as humans, we also inaccurately assess this situation when faced with it as well. In an effort to go one step further, despite this occurrence having no relation to a paradox in the true sense of the word, we have named it so because our minds have difficulty comprehending it at first glance. What do you even mean by “paradox”? The question at hand is merely one of the simple probabilities that harbor no conflicting logic in its rationale.
Any 8th grader who’s learned math class probability is more than capable of understanding how this “birthday paradox” works.
Below is how Wikipedia has explained the question as not being a paradox:
This is not a paradox in the sense of leading to a logical contradiction; it is a paradox in the sense that it is a mathematical truth that contradicts common intuition.
Well then, why is this phenomenon so widespread? The answer is straightforward. The matter is one that is straight out of life itself; in my life of 35 years, I’ve observed that only bright mathematicians call it the “Birthday Problem.” That is why the aforementioned problem is best approached in the sense of a mathematical calculation. Furthermore, having a piecemeal approach is a healthy one. Let’s get started with the probability of two randomly-selected individuals sharing the same birthday.
Before we get into mathematics, we should establish two assumptions. First, we shall assume that February always has 28 days, hence giving us 365 days a year. You can be sure that even if we considered February as having 29 days each year, it would not make a significant difference in the results. We are also going to assume a constant probability for an individual to have been born on any day of the year. So we will be ignoring claims such as there being a greater percentage of births during the summer. Even if we don’t ignore it and if there is no equal probability for every day of the year, the probability of any two randomly-selected individuals sharing the same birthday will be greater than 1/365. This is similar to having two biased dies, where the chance of the dies landing on 2 is higher than any other number. If you roll those two dies, the independent probability of these two dies landing on two is greater than 1/6.So the general problem is quite simple: In a meeting attended by N number of people, what is the probability of at least two people sharing the same birthday?If there are 365 days in a year, to positively state that two people share the same birthday, we need 366 people. That is because though almost unbelievably slim, the chance of 365 people being born on different days still exists. But the 366th person must share a birthday with someone else. If there are 731 people, at least three people must have the same birthday. If there are 1096 people, it will be at least four people, so on and so forth.
But that is not the core of the problem. We set out to figure out the probability of sharing a birthday with someone we’ve just met. Since you are reading this article, let’s figure out the probability of you and I share a birthday. Let’s say that you were born on date X. My probability of having been born on date X is 1/365. Then the probability of me not having been born on date X has to be 364/365. We know that the mathematical probability of something happening or not happening has to add up to 1.
In mathematics, something either happens, or it doesn’t.
As a percentage, 364/365 is equal to 0.99726. So logically, the probability of you sharing a birthday with someone you meet is three thousandths. You might have gotten caught up on 364/365, to which there is a simple explanation: Someone you meet has 364 options for not being born on the same day.
If you meet two new people, your probability of sharing a birthday with one of them will still be very unlikely. However, we now assume that the probability is slightly greater. We know that the probability of not having the same birthday as the first person is 364/365. The probability of not having the same birthday as the second birthday is 363/365 because 363 possibilities are remaining for the third individual. We also desire a case where the other two individuals are not born on the same day. If we multiply these two probabilities in our calculations, we will wind the probability of you not sharing a birthday with those two people whom you have just met. We then get 364/365 x 363/365, which equals 0.9917. So your likelihood of sharing a birthday with one of the two individuals is eight thousandths.
If you meet three new people, the probability of you not sharing a birthday with any of them would be 364/365 x 363/365 x 362/365, which would come out to 0.9836 sixteen thousandths of 1.6%.
We can see the trend as we increase the number of individuals one by one. Because our brains are programmed to think in a linear fashion, we tend to search for answers looking for direct relationships. That is why we must be sure-footed. Below, I have listed the probability of at least two individuals sharing a birthday for the indicated number of individuals in a given instant using a simple algorithm.
If there are two people, then the probability is 0,002739727
If there are three people, then the probability is 0,008204162
If there are four people, then the probability is 0,01635593
If there are five people, then the probability is 0,02713561
If there are six people, then the probability is 0,04046249
If there are seven people, then the probability is 0,05623573
If there are eight people, then the probability is 0,07433534
If there are nine people, then the probability is 0,09462386
If there are ten people, then the probability is 0,1169482
If there are 15 people, then the probability is 0,2529014
If there are 20 people, then the probability is 0,4114385
If there are 22 people, then the probability is 0,4756954
If there are 23 people, then the probability is 0,5072973
If there are 25 people, then the probability is 0,5686998
If there are 30 people, then the probability is 0,7063163
If there are 35 people, then the probability is 0,8143833
If there are 40 people, then the probability is 0,8912318
If there are 50 people, then the probability is 0,9703736
If there are 60 people, then the probability is 0,9941227
If there are 70 people, then the probability is 0,9991596
If there are 80 people, then the probability is 0,9999143
If there are 90 people, then the probability is 0,9999939
If there are 100 people, then the probability is 0,9999997
If there are 103 people, then the probability is 0,9999999
The statistics show the increase in probability alongside the increase in the number of individuals. Yet this increase is not linear. According to the data, if there are 23 people present, the probability of two of them being born on the same day is higher than 50%. Also, if there are 60 people, that probability is closer to 99%. In fact, in a group of 40–50 people, if you were to bet with your friends on there being at least two people born on the same day, you would most likely not lose that bet.
In this problem, the number 23 has a special significance. If you are also interested in soccer, you might know that teams participating in the World Cup must nominate 23 players to join and report it to FIFA. That is why soccer teams can serve as real-life case-studies for the mathematical example discussed earlier.
I should also mention that in the 2014 World Cup final, Germany had beat Argentina 1–0. I must state that the referee was biased against Argentina. I was so upset that I deserved Argentinian citizenship, but that’s beside the point.
In a given setting with 23 people, the probability of at least two sharing a birthday is higher than 50%. So in the World Cup, where there are 32 teams with 23 players each, we can expect that there must be at least 16 teams with two or more players who share a birthday. According to a detailed study below, you will see that exactly 16 teams had players that shared a birthday.
The number of countries with shared birthdays in 2018 is 16 out of 32 teams.1- Russia -> Aleksei Miranchuk & Anton Miranchuk2- Iran -> Saman Ghoddos & Pejman Montazeri3- Morocco -> Yassine Bounou & Ahmed Reda Tagnaouti, Younès Belhanda & Nabil Dirar4- Portugal -> Manuel Fernandes & Cristiano Ronaldo, Raphaël Guerreiro & José Fonte, João Moutinho & Bruno Fernandes5- Spain -> Koke & David Silva6- Australia -> Aziz Behich & Tom Rogic7- France -> Benjamin Pavard & Steve Mandanda8- Peru -> Jefferson Farfán & Nilson Loyola9- Crotia -> Mateo Kovačić & Marko Pjaca10 - Nigeria -> Wilfred Ndidi & Tyronne Ebuehi11- Brazil -> Alisson Becker & Roberto Firmino, Filipe Luís & Willian12- Costa Rica -> Bryan Oviedo & David Guzmán13- Germany -> Niklas Süle & Jérôme Boateng14- South Korea -> Koo Ja-cheol & Kim Young-gwon15- England -> Kyle Walker & John Stones16- Poland -> Wojciech Szczęsny & Łukasz Fabiański, Michał Pazdan & Jacek Góralski, Grzegorz Krychowiak & Rafał Kurzawa, Łukasz Teodorczyk & Łukasz PiszczekSource: Full list of shared birthdays in the last 4 World Cups.
In summary, if you want to impress someone and enter a bet, pay attention to the size of the group you’re in. You could find two people who share a birthday in a seemingly small group (say, 35 people).
Finally, this problem carries a significant role in the field of cryptography. WEP encryption systems that are necessary for the security of wireless networks rely on this model. The logic behind it is that, in a universal set that has “u” number of objects, if you were to independently select two from “n” number of variables that have equal distributions and have the probability of them being equal to each other, be equal to or greater than 50%, the “n” we select must be 1.2 times greater than the square root of the total number of objects in the universal series “u” (n = 1.2 x sqrt(u)). So when we observe the example of 128-bit strings consisting of 264, we find that at least two have a very high probability of being equal to each other.
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