Suppose I am a middle school mathematics teacher enthusiastically lecturing my students about “prime numbers.” After explaining the definition and related properties, just as I am about to continue, a mischievous student asks, “Teacher, can you give an example of a three-digit prime number?” However, I don’t have a prime table up to 1,000 on hand, nor have I memorized primes beyond 100. What should I do? I have no choice but 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 prime! Then a student wonders: how could it be such a coincidence?
The Probability of Primes
The first question is: if you write a random three-digit number, what is the probability that it is prime? There are 143 three-digit prime numbers and 900 three-digit numbers in total. Thus, the probability should be 143/900, which is approximately 1/6. This seems quite low, and “guessing” correctly doesn’t seem easy.
However, in the scenario described above, I am inclined to write down a prime number. Therefore, I certainly won’t write an even number, nor a multiple of 5. Multiples of 3 are also easy to exclude; one just needs to sum the digits and see if the sum is divisible by 3. After excluding multiples of 2, 3, and 5, the remaining numbers are approximately: 900 \times \left(1 - \frac{1}{2}\right) \times \left(1 - \frac{1}{3}\right) \times \left(1 - \frac{1}{5}\right) = 240 In other words, if I casually write a three-digit number hoping it is prime (but without being certain), my pool of candidate numbers is only 240, not 900. At this point, the probability is: \frac{143}{240} \approx 0.6 In fact, it goes further. Multiples of 11 can be excluded quite easily, so the 240 should be multiplied by 10/11. Furthermore, mental division of a three-digit number by a single digit is not complicated, so I would also exclude multiples of 7. Therefore, the actual candidate numbers are only: 240 \times \left(1 - \frac{1}{7}\right) \times \left(1 - \frac{1}{11}\right) \approx 187 Thus, the probability of me guessing a prime number is: \frac{143}{187} \approx 0.76 The probability is about 3/4, so the likelihood of me writing a prime number is very high. Additionally, there is a hidden trick: I might write smaller candidate numbers, where the density of primes is slightly higher.
Why Is It So Coincidental?
The simple hypothetical story above tells us that when calculating probabilities, we must fully uncover the hidden conditions. If “low-probability events” occur frequently, there must be certain factors that have changed the probability, or the event itself was never a low-probability one but was merely packaged as such. Of course, packaging can also go the other way—packaging a low-probability event as a high-probability one, which is a common tendency in advertisements for skincare products and health supplements. These two situations are essentially the same.
There are many coincidences in life. For example, when a man and a woman end up together, they might feel they have a special destiny because so many things were “too coincidental.” For instance, if he asks her to a movie and they both 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 coincidence. However, we often feel things are coincidental because our definition of “coincidence” is too broad. Arriving at the cinema at the same time is a coincidence, but “at the same time” always has a range; a ten-second margin might still count as “at the same time.” Moreover, both parties might have agreed beforehand not to be early or late, so arriving together isn’t a particularly low-probability event.
Most importantly, if they didn’t arrive at the same time, but the girl happened to be exactly one minute late, they might still find it coincidental: “It was exactly 60 seconds, not 61 or 59!” Thus, the occurrence of many “coincidences” actually stems from our overly broad definition of what a coincidence is. To a math enthusiast, being late by 59 or 61 seconds might also be a coincidence (because they are prime numbers). Since the definition of “coincidence” in our minds is so broad, it is not surprising that we encounter them frequently.
Life does not lack coincidence, but the eyes to discover it.
Of course, such an analysis often feels too rational—rational to the point of being unnecessary. For two people in love, for example, they are usually more willing to treat these as genuine coincidences, as in their hearts, this is the manifestation of “fate.” Therefore, in such cases, the best approach is not to try to explain the coincidences, but to let them come even more fiercely.
Finally, let us quote a story from Richard Feynman’s book, Surely You’re Joking, Mr. Feynman!, which describes how Feynman attempted to find a rational explanation for a “supernatural phenomenon”:
Arline died a few hours after I got there. A nurse came in to fill out the death certificate, and left. I stayed with Arline for a while, and I noticed her clock, which I had given her seven years before, when she first got sick with tuberculosis. It was a digital clock, quite a sophisticated thing for those days, which worked mechanically and showed the numbers. It was very delicate and often stopped working, and I had to fix it many times over the years; but I kept it. Now it had stopped again—at 9:22, the exact time recorded on the death certificate!
During Arline’s illness, she had kept that clock by her bed, and it stopped at the very moment she died. I understood that people who are superstitious about such things would not immediately investigate the truth in such a situation; they would assume no one had touched the clock and that the event was unexplainable; and the clock indeed stopped, which could certainly be counted as a startling supernatural case.
However, I noticed that the light in the room was very dim. I even remembered the nurse picking up the clock and turning it toward the light to see the time more clearly, which could easily have caused it to stop.
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