How Could It Be Such a Coincidence! The Hidden Information Behind It

By 苏剑林 | January 21, 2015

Suppose I am a middle school mathematics teacher, excitedly lecturing my students about "prime numbers." After explaining the definition and related properties, just as I am about to continue, a mischievous student interrupts: "Teacher, can you give an example of a three-digit prime number?" However, I don't have a table of primes under 1,000 at hand, nor have I memorized primes beyond 100. What should I do? I am forced to write a few three-digit numbers on the blackboard, such as $173$, $211$, and $463$, and then tell the students, "Let's check if these numbers are prime." The final result: they are all primes! Then a student asks: How could it be such a coincidence?

Probability of Primes

First, the question is: if you write any three-digit number at random, what is the probability that it is prime? There are $143$ prime numbers among the $900$ three-digit numbers. Thus, the probability should be $143/900$, which is approximately one-sixth. This seems quite low; "guessing right" doesn't seem easy. However, in the scenario described, since I intend to write a prime number, I certainly won't write an even number or a multiple of $5$. Multiples of $3$ are also easy to exclude—just check if the sum of the digits is divisible by $3$. After excluding multiples of $2$, $3$, and $5$, the remaining pool of numbers is approximately:

In other words, if I casually write a three-digit number hoping it's prime (but not certain), my pool of candidate numbers is only $240$, not $900$. At this point, the probability is:

Actually, it goes further. Multiples of $11$ can be excluded quite easily, so $240$ should be multiplied by $10/11$. Furthermore, mental division of a three-digit number by a single-digit number is not complicated, so I would likely also exclude multiples of $7$. Therefore, the candidate numbers are actually about:

So, the probability of me guessing a prime is:

The probability is about $3/4$, so the likelihood of me writing a prime is very high. Additionally, a hidden trick is that I tend to write smaller candidate numbers, which further increases the probability of them being prime.

Why Is It Such a Coincidence

This simple hypothetical story teaches us that when calculating probabilities, we must fully uncover hidden conditions. If "low-probability events" happen frequently, there must be factors that have changed the probability, or the event itself isn't low-probability but has been framed as such. Conversely, there is another tendency: to package low-probability events as high-probability ones, common in advertisements for skincare and health products. These two situations are essentially the same.

There are many coincidences in life. For example, when "he" and "she" get together, they feel a special fate because many things seem "too coincidental." For instance, if he asks her to a movie and they happen to arrive at the cinema at the exact same time—isn't that a coincidence? Yes, the probability is quite low, so it seems like a remarkable coincidence. However, our feeling that many things are "too coincidental" often stems from our definition of "coincidence" being too broad. Arriving at the cinema at the same time is a coincidence, but "at the same time" always has a range—an error of ten seconds might still count as "simultaneous." Moreover, they might have both aimed not to be early or late, so arriving together isn't such an extremely low probability.

Most importantly, if they weren't exactly simultaneous but she arrived exactly one minute late, they might still find it coincidental: "Exactly 60 seconds, not 61 or 59!" Many "coincidences" occur because our definition of what constitutes a coincidence is too loose. For a math enthusiast, arriving 59 or 61 seconds late might also be a "coincidence" (because they are primes). Since our brain's definition of "coincidence" is so broad, it's not surprising that we encounter them often.

Life does not lack coincidences; it lacks the eyes to discover them.

Of course, such an analysis can feel too rational—rational to the point of being unnecessary. For a couple in love, they are usually more willing to see these as genuine coincidences, as they represent "fate" in their hearts. In such cases, the best approach is not to attempt to explain the coincidences, but rather to let the coincidences come even more fiercely.

Finally, consider a story from Richard Feynman’s "Surely You're Joking, Mr. Feynman!", where he describes how he sought a rational explanation for a "supernatural phenomenon":

A few hours after I arrived at the hospital, Arline died. A nurse came into the room to fill out the death certificate and then left. I stayed with Arline for a while longer and happened to notice the alarm clock I had given her. That was seven years ago when she first contracted tuberculosis. In those days, this digital clock was quite sophisticated; it worked on mechanical principles and could display digits. Because its structure was so delicate, it broke easily, and I had to fix it every so often; yet I never threw it away. This time it had stopped again—at 9:22, exactly the time recorded on the death certificate!

While Arline was sick, she always kept that clock by her bed, and it stopped at the very moment she passed away. I knew that people who are inclined toward such things would, in this situation, not immediately investigate the truth; they would assume no one had touched the clock and that the event was inexplicable. The clock had indeed stopped, and it could certainly be counted as a startling supernatural case.

However, I noticed that the room's light was very dim. I even remembered the nurse picking up the clock and holding it up to the light to see it better, which could easily have caused it to stop.

Original URL: https://kexue.fm/archives/3210

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Category: Everything | Tags: Probability, Analysis

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