Could Statistics and Maths Explain Coincidences?

I’m pretty sure you’ve heard people overthink events that occurred in their lives, accrediting them to supernatural causes, horror-related reasons, or anything that’s more than just “a coincidence.” 

Things like these are certainly common, ranging from the most basic, harmless “oh that classmate of mine has the same birthday as I do” to the extreme cases of “that person is literally copying whatever I do.” While the causes of these issues vary from one another, there’s one unifying concept that one has to remember when dealing with these seemingly eccentric events yourself, or when a friend of yours talks about this: probability.

Perhaps you’ve also heard the phrase “nothing is coincidental” but if you really think about it, maybe some things are in fact expected to occur, thanks to probability. There are probability rules which clearly show that certain events are predictable – there are statistical results if you measure enough of them, hence negating the initial phrase.

 

Statistics

Why so? Well, according to statistics, the expected number of times an event happens could be calculated. When tossing a coin, the probability of getting heads is equal to the probability of getting tails: 0.5. If you toss the coin 10 times, the expected number of times you’re going to get heads in those 10 times is 5 – the product of multiplying the number of trials and the probability of getting heads.

Two 20kr gold coins from the time of the Scandinavian Monetary Union. Anonimski, CC0, via Wikimedia Commons.

Of course, you might not get exactly 5 heads in your 10 trials, since these values are just that: expected. However, if you try and toss for an even greater number of trials, say 100 times or 1000 times, then you’re bound to getting roughly 50 heads and 500 heads respectively – that is if you have enough will to perform as many flips and record each of their results.

If anything, the fact that you have to collect a large amount of data to draw such a conclusion highlights a fundamental requirement of the study of statistics and probability. On a small scale, events may be unpredictable, or more than just a coincidence. Yet, if you dive into the realm of the vast, where you calculate probabilities on the larger scale, you see that they form a general trend, where even the most unexplainable, minor details are indeed calculatable.

Hack, even the definition of statistics according to Oxford Languages is “the practice or science of collecting and analyzing numerical data in large quantities,” so it shouldn’t be surprising that to learn the statistical representation of an event, one must gather large amounts of data before exploring trends hidden between them.

Certainly not every data can be gathered in large amounts, perhaps due to cost, time, or practical issues that lie underneath. Nonetheless, one can always calculate probabilities in the realm of the theoretical, assuming that the laws of probability prevail under all circumstances. With such means, even the most unintuitive events can happen and be proven using the study of probability. 

 

Birthday Paradox

Enter birthday paradox or sometimes also called the birthday problem. This paradox sounds unintuitive at first glance, and it is reasonable to think so. Wikipedia presents the problem in this way:

In a set of n randomly chosen people, some pair of them will have the same birthday. By the pigeonhole principle, the probability reaches 100% when the number of people reaches 367 […]. However, 99.9% probability is reached with just 70 people, and 50% probability with 23 people. These conclusions are based on the assumption that each day of the year […] is equally probable for a birthday.

To reiterate, the birthday problem imposes that in a random group of at least 23 people, there is a 50% chance that two people in the group share the same birthday. The chance of this event of birthday sharing happening skyrockets to almost being certain to happen, with a 99.9% chance when there’s merely 70 people.

 

How come, one may ask? While there are 365 days in a year, you would only need 23 random people to have two of them share the same birthday with a 50% chance! There are so many days in a year, yet so little few people are required to have such an event happen! Does this not sound unintuitive?

Well, for one thing, the mathematics speaks for itself. In my opinion, the birthday problem needs no complex ideas to be proven – so let’s hop into it. 

 

Joint Probability Distribution

Firstly, the concept of a joint probability distribution has to be laid out in order to make sense of this paradox. Despite its scary-sounding name, it’s quite simple to think about. Imagine Bob has two coins, and he flips them both at the same time. What is the probability that his two coins will both land on heads?

Before showing the mathematical proof of the probability, it’s quite clear that there are only four possible outcomes of the two coins, where the two coins can land on:

  1. Heads and Heads
  2. Heads and Tails
  3. Tails and Heads
  4. Tails and Tails

Therefore, the probability of the two coins landing on heads is one in four, or 0.25. There are only four possible outcomes and the combination of two heads is one of those four. Mathematically, this is represented with the joint probability distribution. Each coin has a 0.5 probability of landing on heads, and to calculate the probability that both with land on heads is quite simple: you multiply the probability of each coin landing on heads.

P(A \cap B) = P(A)\times P(B)

That means, the probability of having both coins land on heads is given by 0.5 times 0.5, equalling 0.25, which we have also attained earlier. And that’s joint probability distribution! The probability of two independent events happening consecutively is the product of each of the probability of the event. You can always extend this idea to more than two events, say n times, hence you can simply multiply those n independent probabilities.

 

Complement Rule

Then, to ready ourselves for the birthday problem, you would need to understand how events can be complementary to each other, given that they are mutually exclusive. Disregard these fancy words and allow me to put it this way:

Some events are mutually exclusive with each other, meaning that if one of them happens, then the other would not, nor would it both happen simultaneously. For example, Bob tosses a coin. The event of getting heads is mutually exclusive to getting tails since neither of those can happen simultaneously: if Bobs gets heads, he does not get tails, the same goes vice versa.

Puzzle pieces, black and white. Willi Heidelbach, CC BY 2.0, via Wikimedia Commons.

Putting it mathematically, Bob can calculate the probability of getting tails simply from knowing the probability of getting heads. That is, the probability of getting tails is given by 1 minus the probability of getting heads since the two events are complementary to each other.

Of course, this may seem trivial to calculate and a waste of time if you do so with only two possible outcomes. However, you can always extend this idea to cases where there are more than two possible outcomes – like the two-coin toss problem imposed by Bob.

We know that the probability of getting two heads is 0.25, or one in four chance. What if we’d like to calculate the probability of not getting two heads? Well, since the event of getting two heads is mutually exclusive to every other possible outcome, the probability of not getting two heads is given by 1 minus 0.25 (the probability of getting two heads), which equates to 0.75. Meaning, in three out of four chances, you will not get two heads.

P(A')=1-P(A)

 

Proof of Birthday Paradox

With all these foundational concepts laid out, you’re ready to tackle the birthday paradox – with the particular case of 23 random people. Rather than calculating the probability of two random people sharing the same birthday, let’s instead calculate for the complement of that event: the probability of two people not sharing the same birthday, using the knowledge we’ve earned earlier.

Let’s say Bob is in a group of 23 people. The probability that Bob has a birthday is, of course, 365 out of 365. Out of the 365 days in a year, Bob has a 1.0 chance that he has a birthday (since he is lives). Then, for the next 22 people, they should not share a birthday with Bob (remember, we’re calculating for the complement).

Say that Alice is in the group, together with Bob. The probability that Alice does not share her birthday with Bob is 364 out of 365, since there’s that 1 particular day when it’s Bob’s birthday, and we do not want that. 

After Alice, say Carlos is the next person in line. The probability that Carlos does not share the same birthday with Bob nor Alice is 363 out of 365, as the subtracted two days belong to Alice’s and Bob’s birthday respectively (which the three of them do not share).

Continue such process for all 23 people, and the math leaves:

P(A') = \frac{365}{365}\times\frac{364}{365}\times\frac{363}{365}\times\frac{362}{365}\times \cdots\times\frac{343}{365}

Equivalently, its factorized form:

P(A') = \left(\frac{1}{365}\right)^{23} \times (365 \times 364 \times 363\times \cdots\times 343)

Notice that the term 23 corresponds to the number of people in the group, and 343 is equal to 365 minus 22 (one-off from the number of people in the group). If you calculate the probability of the complement, you’ll find out that it results in about 0.492703 chance.

P(A')\approx0.492703

Due to the complement rule, the probability that two people share the same birthday in the group of 23 people is 1 minus 0.492703 which is about 0.507297, hence the 50% chance we’ve presented earlier!

P(A) \approx 1-0.492703 \approx 0.507297

You can modify such a process with 70 people, and find out that it is consistent with what we have shown earlier, that is, 99.9% chance of two people sharing the same birthday.

 

Implications of Birthday Paradox

What could we make of such a paradox, where so few people are required to find a form of collision – where in this case the collision is their birthdays? Doesn’t this unintuitiveness conflict with the way of thinking whereby one is insanely unique in the vast amount of worldly population?

Well, as we have gone through all the maths, collisions are inevitable. Despite a large number of possible outcomes, you would only need very little to share something in common with another person. Remember, collision stems merely from mathematics, not some kind of wizardry requiring the interpretation of a higher-power being.

I’m sure you or your acquaintance has faced such an event in life, where say, you and your friend somehow bought the same type of bottle, with the same color, of the same size, from the same brand. Does this mean both of you are somehow a match? As if the gods planned all of this to happen and it’s more than just a coincidence? Think again.

Spider Man pointing at Spider Man.

In a world as huge as ours, yet with a population as dense as there is in this world, would you be surprised if a person living across the globe from you share similar traits? Or share the same likes and dislike? Or even look similar to you? Now that you’ve understood how collisions are very much likely, perhaps inevitable, events like are not surprising after all…

If we were to find collisions, are we entitled to cherry-pick features that somehow we share, aside from the ones that actually collided? If, say, Bob and Alice share the same birthday, what other common factors do they share which may lead to such collisions, maybe their parents are somehow related? Maybe their parents’ doctors are siblings who somehow conspire to have their patients’ child share the same birthday?

Of course not! It’s very silly to connect too many dots when there are, in reality, only two which collide with each other! And this trait is found in many generations, ranging from extreme ones like believing in conspiracy theory, to the most harmless way of interpreting events. Like it or not, however, you’re going to find such people who try and connect way too many dots, such that they have a reasonable explanation behind collisions and reasons that could be pulling the strings behind the birthday paradox.

 

Case: The Problem With Seraphine

Enter “The Problem With Seraphine,” an article written by an author who essentially believes that a fictional character, League of Legends’ Seraphine, is based on her. “I think she’s based on me,” said author Stephanie.

Stephanie provided a short version of “pieces of evidence” for her case, in which she wrote:

The short version is that a lot of details about Seraphine line up very closely with facts about myself: her name, her drawings, her cat, a lot of her pictures, her hair color, her eye color, her face shape, and even where she’s from.

Moreover, Stephanie argued that aside from visual similarities, she in fact dated a Riot Games (developers of League of Legends) employee, who’s actually worked on League characters just like Seraphine. You can read further details about her and her date’s interactions in her Medium article.

That’s Stephanie & Cat on the left, and Seraphine & Cat on the right. Stephanie’s Medium.

Basically, Stephanie provided seemingly strong shreds of evidence that the fictional character Stephanie is based on her, and she’s uncomfortable with that. Such a collision occur, implied by Stephanie’s arguments, could highly be due to the fact that her former date copied her characteristics: looks, likes, possessions, places, etc.

Allow me to express my take on the case. This issue is an overly exaggerated form of interpreting a collision. In a world where possibly millions of other girls also dated a Riot employee, and maybe have their traits embodied into a fictional character, why would Stephanie’s case be unique?

Yes, Seraphine and Stephanie do share similar traits. Yes, Stephanie did date a Riot employee who worked on in-game champion skins. But does it all necessarily sum up to be some form of conspiracy? A form of planned plagiarism based on a former date? With all due respect, I do not agree with that.

Say Alice and Bob share the same birthday, and maybe somehow their parents’ name starts with the same letters, grew up in the same country, surprisingly went to the same high school, what does that imply? Absolutely nothing!

Again, collisions are inevitable, especially in the global population of 7.8 billion people as of the time of writing. Stephanie might not be the only girl who experienced the same case, perhaps there are other girls, who might just not acknowledge such an event, or even care in the first place.

Although the debate of whether or not Riot is making billions of profits from a real-life person’s characteristics is ethical or not, the point is this: Stephanie’s case with Seraphine is not unique to herself. Collisions, or shared traits, are bound to happen and shouldn’t be interpreted as if it’s a conspiracy – it is just math.

 

Closing Remarks

We have gone through a lot of different concepts and ideas in this article, starting with the conceptual understanding of statistics, proof and implications of the birthday paradox, and finally ending the discussion with a case of collision, as well as its false interpretation. 

Understand that collisions: sharing of birthdays, traits, characteristics, looks, visuals, details, maybe simply that: a collision. Though certainly there may be influences from the so-claimed “authentic” or “original” source, sometimes people just get the same ideas and work independently of each other.

Arch-rivals Isaac Newton and Gottfried Leibniz famously fought over the credit for inventing calculus. Arstechnica.

Isaac Newton and Gottfried Wilhelm Leibniz both invented Calculus, independently of each other. Newton was in England, while Leibniz was in the Roman Empire. Does that mean the two of them were copying each others’ work, or maybe share the same ghosts who whisper to them the secret sauce behind the invention of Calculus? 

This Calculus controversy as to who invented Calculus first went on and on and supporters of the respective figure battled it out without end. Perhaps they should have known that events like such are bound to happen and that it would require so little population to get collisions as such. In the end, it’s not the matter of who invented Calculus first, it’s about leveraging the tools that humankind has worked hard for. Don’t mind the collision, it’s just a coincidence.

 

Featured Image by Andrew Gray — original photos, Alexey Gomankov — collage, CC BY-SA 3.0, via Wikimedia Commons.

Wilson Wongso Written by:

Indonesian Computer Science Student, Private Tutor, and Content Writer. At times I read, game, or watch. My boss calls me Jack of All Trades; I could not agree more.

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