"Hey, you're saying it's a coincidence!" We usually talk about this sentence. Last Tuesday, I took the subway to work, and as soon as I entered the car, I bumped into a high school tablemate who I hadn't contacted for ten years, and the phone cases of the two were still the same cherry blossom pattern.

Everyone has encountered this kind of life vignette more or less - watching a short video suddenly heard the ringtone of the mobile phone set by yourself, and the mantissa of the takeaway order number is just the last four digits of the ID card, these coincidences are like colorful jelly beans sprinkled in life, adding some fun to ordinary days. But some coincidences are so outrageous that people doubt life.

美國德州"彩票女王"Joan Ginter（之前誤寫作John Gainer），這位姐姐堪稱現實版錦鯉：1993年喜提500萬美元大獎后，2006年再中200萬，08年300萬，10年直接斬獲1000萬！這劇情連編劇都不敢寫，怕被觀眾罵太假。當時媒體報導說這概率相當於18後面跟24個零分之一，比連續四年被雷劈還誇張。

Today, let's peel off the lucky coat of these "chosen ones" and see what anti-common sense truths are hidden behind probabilities.

[When Probability Meets Reality: Don't Be Fooled by the Number Game]

先來算算這位錦鯉姐姐的"歐皇指數"。按德州樂透54選6的規則，中頭獎概率是2580萬分之一。如果簡單粗暴地四次方計算，確實會得出44×10²⁸分之一的恐怖數位——相當於把全地球的沙子一粒粒數完1000遍，期間每次都能精準找到標記過的那粒沙。

But there is a fatal flaw in this calculation: just like insisting that a star is "one in a million in beauty", the premise is that he really stands out from the audition of millions of people. In fact, the algorithm of the media corresponds to "specifying that someone must win the lottery four times in a row", while we need to calculate the probability of "at least one four-time winner among the global lottery players", which is completely different.

For example, if you have 27 people in your circle of friends, the probability of meeting a birthday on the same day is as high as 0%. But to say "it must be the same as the designated person's birthday", the probability immediately plummets to 0.0%. This is the magic of conditional probability: the preconditions are different, and the outcomes are very different.

[The Magic of Conditional Probability: This is how koi fish are raised]

假設全球有1億忠實彩民，每人每周買10注堅持20年，累計人均買彩1萬次。用泊松分佈計算就會發現，出現至少一個四冠王的概率其實有1/2500萬。雖然還是小概率，但比之前的天文數位合理多了——相當於中500萬的概率，突然變成中輛寶馬3系的程度。

Not to mention that the variables in reality can't stop at all:

1. 中獎后本金暴漲：第一次中500萬後，買彩票可能從機選1注變成包100個號
2. Play upgrade: You may switch to the higher prize Powerball or EuroMillions
3. The media spotlight effect: there may still be unreported winners, but we don't know
4. System vulnerabilities: In the early years, lottery anti-counterfeiting technology was limited, and it is not excluded that there are loopholes in the technical flow

就像你突然發現鄰居中了雙色球，卻不知道人家其實每期都買200塊複式票。所以表面上的"天選之子"，可能只是把1/100萬的概率，通過持續投入提升到了1/1000。

Even the so-called "once-in-a-century" rainstorm is now on the hot search every three or five times. This is not God's convulsions, but the statistical caliber is at work - the Meteorological Bureau said that "once in a century" refers to a specific region, but there are more than 3000 prefecture-level cities in the country, and each city has more than a dozen types of disasters such as high temperature/heavy rain/hail, which is equivalent to 0 lottery chances per day, and winning the lottery is naturally not unusual.

The next time you encounter a miraculous coincidence, remember to activate the "Human Sober Mode":

1. Confirm the base probability: is it 10/0 or 0/0 billion?
2. Calculate the total sample size: the number of participants * the number of attempts?
3. Check Independence: Does the Previous Result Affect the Subsequent Probability?
4. Be wary of survivorship bias: How many of these incidents go unreported?

[Summary: Dancing between certainty and serendipity]

In the final analysis, human obsession with coincidences stems from our eternal quest for certainty. In ancient times, astrological divination was used to explain coincidence, and modern probabilities and statistics were used to decipher metaphysics, which essentially wanted to put the reins on randomness. The Texas lady who won the jackpot four times was later revealed to be a frequent visitor to the local lottery shop, and after winning the lottery, she even regarded the lottery as an investment. You see, there is no such thing as good luck from heaven, but it is a game of probability when it meets a persistent player.

Just like the subway mentioned at the beginning, if you know that the other party lives in the same area as you, works in a similar industry, and the commuting time coincides, this coincidence is instantly reasonable. In the large-scale reality show of life, the real miracle may not be those small probability events, but that we will always retain our curiosity and exploration of the unknown - after all, being able to salvage the pearl of certainty in this sea of probability is the most moving superpower of human beings.

[Text source: Dr. @Kai's video content]