Why is there no winner in long-term gambling? You need to understand the "Gambler Loses All Principle"

Why not gamble? This question needs to be discussed every year and every month. I previously made a video titled "Why there are no winners in long-term gambling". This time I converted the video into a text version for everyone to review.

We often hear the saying: There is no winner in the long run. Even if it is a seemingly fair gambling game, as long as the gambler continues to gamble for a long time, he will definitely go bankrupt. Do you know why?

Let's look at an example: suppose there is a fair gambling game, and in each round, the gambler has a 50% chance of winning 1 yuan and a 50% chance of losing 1 yuan. The gambler originally has A yuan, and he will quit in two situations: either losing all his money or winning B yuan. What is the probability that he will eventually lose all his capital and leave?

We can use an image to describe this problem. There is a number line, and the gambler is at position A. He will randomly move one square to the left or right each time. If he moves to position 0 on the left or position B on the right, the game ends. So what is the probability that the gambler will eventually move to position 0 to end the game?

In each game, the gambler randomly moves one square to the left or right.

It is not difficult to solve this problem: suppose a gambler has n yuan, the probability of losing everything is P(n).

The correspondence between the gambler's original capital n and the probability of losing all P(n)

According to the rules of the game, if n=0, the gambler loses all and leaves the game, the probability is

P(0)=100%

If the gambler has B yuan, he will leave the market satisfied and will not lose again. Therefore, the probability of leaving the market with all the money lost is

P(B)=0

In each game, the gambler randomly wins or loses 1 yuan, that is, the gambler's money n has a 50% chance of becoming n+1, and a 50% chance of becoming n-1, so:

P(n)=50%P(n+1)+50%P(n-1)

By multiplying both sides of this formula by 2 and then moving the terms, we can easily get:

P(n+1)-P(n)=P(n)-P(n-1)

You will find that the difference between two consecutive terms of the P(n) sequence remains unchanged. This is an arithmetic sequence! Moreover, its first term P(0)=100%, and the last term P(B)=0. It is a gradually decreasing arithmetic sequence, and each term is 1/B less than the previous term.

The relationship between a gambler's target capital and the probability of losing everything

We can draw a graph of the relationship between the probability of losing everything P(n) and the current amount of funds n. Using the proportional relationship, it is easy to calculate the probability of a gambler losing everything when his funds n=A.

P(A)=1-A/B

That is, the probability that a gambler loses everything is equal to 1 minus the gambler's original money A divided by his target B.

We can discuss this result: If the gambler has 100 yuan, that is, A=100

If the gambler hopes to win 120 yuan and then quit,

B=120, P=1-100/120=1/6

This means that there is a 1 in 6 chance that the gambler will lose everything;

If the gambler hopes to win 200 yuan before quitting,

B=200, P=1-100/200=1/2

This means that there is a 1/2 chance that the gambler will lose everything;

If the gambler wants to win 1,000 yuan before quitting,

B = 1000, P = 1-100/1000 = 9/10

This means that there is a 9/10 chance that the gambler will lose everything;

You will find that the bigger the gambler's goal, the greater the probability of losing everything. What if he keeps gambling and never quits no matter how much money he wins? At this time, the goal B becomes infinite (B=∞), so the probability of losing everything is

P=1-100/∞=100%

This means that if a gambler continues to gamble, he will definitely lose all his money, and there will be no winner in the long run!

In the process of gambling between gamblers and casino owners, even if it is a fair game, since the amount of funds of the casino is far greater than that of the gambler, it is almost impossible for the gambler to win the casino into bankruptcy, and the gambler will eventually lose everything and leave.

Similarly, when trading stocks, if your goal is just to make 10%, it is relatively easy. But if you want to double your money, there is a 50% chance that you will not achieve your goal. If you increase your leverage to 10 times through margin trading or options and futures, then you will have a high probability of losing all your money.

Pushkin, a great Russian poet, wrote a fairy tale called "The Fisherman and the Goldfish": A fisherman saved a magical goldfish, which fulfilled many of his wishes. However, the fisherman's wife was always dissatisfied, and in the end, the goldfish took away everything he had given her, and the couple returned to the shabby house where they first lived.

Fisherman and Goldfish

The moral of this story:

A greedy person will end up with nothing.

Source: Mr. Li Yongle