Let’s talk about it from a mathematical perspective: why do people who are addicted to gambling always end up going bankrupt?

In the recent hit drama "Crazy", there is a scene:

Gao Qiqiang, a big boss, gave his subordinate Tang Xiaolong a casino and told him: These are the most popular machines nowadays, and the odds can be set as you like, 20%, 30% or even higher. Tang Xiaolong made a lot of money in a short period of time with these gambling machines.

Some people make a fortune, while others lose everything.

Today, we will talk about how casino dealers make money from gamblers from a mathematical perspective. If you understand this principle, you will understand why people who are addicted to gambling will eventually go bankrupt.

Image source: Tuchong Creative

The law of large numbers determines that the dealer always has an advantage over ordinary players

The games in the casino are designed by the casino dealer. When designing each gambling game, the dealer will definitely have a slight advantage over ordinary players in terms of probability.

Let's take roulette as an example (see Figure 12-1). The rules of roulette are very simple: a turntable is divided into 38 squares, and the player guesses which square the ball will stop at. If the guess is correct, the casino usually pays the player at a ratio of 35:1. In other words, if you bet 1 yuan, if you bet correctly, you will not only get back the 1 yuan, but the dealer will also give you another 35 yuan; if you bet wrong, you will lose the 1 yuan you bet.

Because there are 38 grids, the probability that the player guesses which grid the ball will fall into is 1/38. Probability is a mathematical concept. In order to explain in detail what the probability of "1/38" means, we assume that a player plays the game many times and then counts whether his guesses are correct.

Because the player either guesses "right" or "wrong" each time, we simply arrange the player's "right" and "wrong" answers each time, and the final result may be like this:

Wrong, right, wrong, wrong, wrong, right, wrong, wrong, wrong, wrong, wrong, wrong, wrong, wrong, wrong

Wrong, wrong, right, wrong, right, wrong, wrong, wrong, wrong, wrong, wrong, wrong, wrong, wrong, wrong

Wrong, wrong, wrong, wrong, wrong, wrong, wrong, wrong, wrong, wrong, wrong, wrong, wrong, right

Wrong, wrong, wrong, wrong, wrong, wrong, wrong, wrong, wrong, wrong, wrong, wrong, wrong, wrong, wrong, wrong

Wrong, wrong...

These results may be only part of the total number of times a player plays the game (for example, 10,000 times), and the percentage of correct answers in these 10,000 times is calculated, it will be very close to 1/38. That is, 10,000×1/38≈263 (times). This is the actual meaning of the probability of "1/38".

Note that the above method actually counts the frequency of guessing "right". That is, if the number of times is large enough, the "frequency of a certain result" is equal to the "probability of the result".

In statistics, there is a term called the "law of large numbers" that explains this phenomenon. The law of large numbers is the cornerstone of statistics. It means that as long as an event occurs enough times, the frequency of its occurrence will be equal to its probability.

We note that the law of large numbers requires the condition of "enough occurrences" to be true. Only when the number of occurrences is large enough will the statistical frequency equal the probability, and the more times it occurs, the closer the statistical frequency is to the probability.

Let's look at the player's earnings when he plays enough times. Assuming he bets 1 yuan each time and bets 10,000 times, then according to probability, the number of times he guesses correctly should be very close to 263 times. Since he gets 36 yuan for each correct guess, his earnings for 10,000 guesses are roughly 263×36 = 9468 yuan.

However, since he invested a total of 10,000 yuan, he lost about 500 yuan in total.

Note that although 500 yuan is not much, it is a stable loss. Because as long as you play enough times, the frequency of guessing correctly will be very close to 1/38. Under this probability, if you bet 1 yuan per game, there is only a 1/38 probability of getting 36 yuan back, so the average loss per game is:

This is the mathematical principle that “long-term gambling will lead to losses”.

We can see that when designing the game, the dealer always makes his chance of winning slightly higher than the player. This advantage is usually very small, 5% to 10%. But don't underestimate this little bit of probability advantage. With this little bit of probability advantage, the dealer increases the number of bets. In this way, according to the law of large numbers, the dealer can make money steadily.

Whether you can catch the doll depends not on skills but on luck

Not only do casinos use the law of large numbers to make money steadily, this idea has also been quickly used by businesses in different industries. Let’s take claw machines as an example.

I also played doll machines when I was in college. Unlike the current doll machines with three claws, the doll machines I played at that time only had two claws. But the advantage is that as long as the two claws grab the doll, they can usually grab the doll out. Therefore, the performance depends largely on the skills of the person playing the doll machine. Experienced people can find the right position to put their claws down, and can often grab a lot of dolls. I remember one night, I played in a mall for an hour and grabbed a bag of dolls.

But in recent years, claw machines have been upgraded. First, the claws have been changed from two to three. But this is not the key point. The most important thing is that the tightening and loosening rules of the claws of the claw machine can be set! For example, the merchant can set the probability of the claws clamping this time to 1/10, which means that on average, for every 10 times of clamping, the claws will loosen when they are raised 9 times. If you have played a claw machine, you will know that if the claws are loose this time, it is almost impossible for you to get the doll out.

This setting of probability is revolutionary. It means that merchants have gotten rid of the shackles of "players' skills" and can play the game with players directly at the level of probability.

If it is set that it costs 2 yuan to play a game, the price of each doll is 10 yuan, and the merchant sets the probability of the claw clamping to 1/10, then the average loss of the player playing a game is 1 yuan. This is also based on the law of large numbers. The more times the player plays, the more the actual situation conforms to this average loss.

We can see that claw machine merchants also take advantage of the "probability advantage" and the "law of large numbers." As long as there are enough people participating, they can always be invincible.

Master the law of large numbers and be your own "banker"

What insights can we get from the examples of casinos and claw machines that are useful for our daily work and life?

First, you need to work hard to improve your basic probability. This is very clear. Basic probability is the core and the key factor to achieve your goal.

Second, if you have a high base probability of accomplishing something, then repetition is your best friend, and you need to repeat it as many times as possible.

For example, you are a self-media person and want to write a hit article. We all know that in many cases, hit articles are hard to come by. Even if your article is of high quality, there is no guarantee that it will become a hit. If your level reaches an average of 100 articles to have 1 hit article, the probability is already very high, so you should write more at this time. As long as your level is up to standard, the law of large numbers will help you.

Third, if your base probability is lower than that of your competitors, then you should think as follows.

First see if you can increase the basic probability. If not (for example, you are a casino player), the best option for you is not to participate in gambling and jump to another game that is in your favor in terms of probability.

It can be seen that mastering mathematical thinking is also of great significance to our daily life in seeking benefits and avoiding harm.

The article is produced by Science Popularization China-Starry Sky Project (Creation and Cultivation). Please indicate the source when reprinting.

Author: Liu Xuefeng, Associate Professor and Doctoral Supervisor at Beijing University of Aeronautics and Astronautics

Reviewer: Deng Qingquan, Associate Professor, School of Mathematics and Statistics, Central China Normal University