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On August 18, 1913, the ball landed on black in the last 10 spins of the red-and-black roulette table at the Monte Carlo Casino in Monaco. People thought that the next spin must fall on red, so gamblers from all over the casino began to bet on red. As the result of every rotation of the wheel is black, the crowd increasingly believes that the next round will be red. But in the end, black appeared 16 times in a row, a total of 26 times in a row, the probability of this happening is about 6600 1/10000. Gamblers lost millions of francs because they fell victim to the gambler's fallacy.

If you were at the Monte Carlo Casino that day, would you bet on red or black? In this article, we will see what went wrong with gamblers and how they can avoid amazing losses with the right thinking. First of all, let's start by flipping a coin.

We all know that when we flip a coin, it has a 50% chance of going up, and it also has a 50% chance of going down. Now, when I flip the coin, it shows the front, and I have flipped the coin three times in a row, all face up. The chance of facing up four times is 1/16 or 6.25%, and if you flip the coin again, many people may think that there is a higher chance of the opposite this time. If so, it falls into the gambler's fallacy that independent past events will affect independent future events in the same randomized trial.

But this is seriously wrong, this coin will not remember the last few flips, and the chance of getting heads or tails in the next flip is still 1/2. Coin flipping is so-called statistically independent, and each event is independent of previous and future events. We tend to think that opportunity is self-correcting, or that there must be a cosmic balance, so we fall into the gambler fallacy.

In fact, this belief in cosmic balance is not as stupid as it sounds. If we flip a coin enough times, we will begin to see something very interesting. In 1939, a South African mathematician named J.E. Kerrich traveled to Europe but ended up in a Danish prison. Out of curiosity or pure boredom, he flipped a coin 10,000 times and recorded the number of times it fell on the front. The results showed that the more times the coin was flipped, the closer the relative frequency of the front became to 50%.

The pace of life of modern people is so fast that they don't have enough free time to flip a coin 10000 times, but we can simulate it with a computer. And we can see the same pattern every time, and the relative frequency of the front is always about 50%. This is not only a feature of a coin, but also a 1/6 or 16.67% chance that a dice will fall on each side. If we roll the dice 10000 times, we will find that the relative frequency of each side is close to 16.67%.

This phenomenon is called the law of large numbers, and the more experiments we do, the closer the relative frequency of the result is to the probability of the event. The probability of the positive side of a single flipped coin is 50%, and after enough flips, the relative frequency of the positive side is about 50%. This gives us a feeling that there seems to be an opportunity for self-correction and that there is some kind of cosmic balance.

Therefore, on the one hand, we are statistically independent, which tells us that each round has nothing to do with the last round, and gamblers mistakenly believe that past results will affect future results. On the other hand, we have the law of large numbers, which tells us that the opportunity system does eventually balance. So it seems right for gamblers to expect red after a long period of black. So how can we reconcile these two seemingly contradictory ideas?

There is a reason why it is called the law of large numbers, and we can see this consistency or balance only after a large number of experiments. Small sample sizes usually show great variability, and it is more likely to see black 26 times in a row. One of the most famous examples of this phenomenon comes from a study done by the Bill Gates Foundation.

The foundation studied the educational results of schools and found that small schools always topped the list, deducing that small schools lead to better education. The foundation applied small school technology to large schools, such as reducing class size and reducing the teacher-student ratio, but these methods failed to produce the huge benefits they had hoped for. Because the foundation ignores a key thing, the schools at the bottom of the list are also primary schools. It is not that the performance of primary schools is better, but that the test scores of primary schools change more. Some prodigies can significantly raise the average in primary schools while in larger schools these extreme scores are incorporated into larger averages with little impact.

The 26 consecutive black on the roulette table is only a relatively short continuity in a series of millions of rotations. The mistake made by gamblers is to regard a small sample size as the same as a large sample size, but in fact the difference between them is so large that such a small sample size cannot obey the law of large numbers.

This article comes from the official account of Wechat: Vientiane experience (ID:UR4351), author: Eugene Wang

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