Can anyone convince a depraved gambler that it's a character flaw. But if you are still a rational person, stop being obsessed with so-called luck. What gamblers can rely on is the blessing of their ancestors, and the bosses behind the casino are such great gods as Gauss, Kelly, and Bernoulli. How could you win the dealer? Gamblers are superstitious about luck, while casinos believe in
mathematics
What if they don’t come?” Ye Han said with a smile: “A gambler once, a gambler for a lifetime, what they’re worried about is what if the casino isn’t here.
” Han was much more rigorous back then, and the casino concentrated his mathematical experience in probability, series, and limits. An ordinary gambler, as long as he gambles for a long time, he will lose his money in the end. Except for Zhou Xingchi in the movie, Zhou Xingchi in reality doesn't believe in all kinds of winning skills.
Gamblers will never understand that what they are betting against is not luck, nor the banker. They are competing with masters of mathematics such as Dirichlet, Bernoulli, Gauss, Nash, and Kelly. What is the chance of winning?
What you can see is the probability, what you can't see is the trap.
Let's talk about the simplest gambling game first: betting on luck and guessing coins.
The rules are like this, toss a coin, heads win and tails lose, you can take away double the money if you win, and lose your principal if you lose, do you want to play? You may think, alas, this game is not bad, fair! It just so happened that luck was good, and the first one won 100 yuan! You are so happy. At this time, the dealer tells you, you see that you have won so much, and I worked hard to set up a game, but in the end I didn’t get anything. Otherwise, if you win, just leave it to me 2%, even if it's the relief brother, give it a thumbs up! When you hear it, 2% is only so little, take it, it’s not bad money! Well, that's it for now.
However, what you never imagined in your dreams is that it is this small 2%, but in the end, you lose everything and ruin your family.
This small winning probability of 2 points seems inconspicuous, but coupled with the "Law of Large Numbers", it becomes a weapon for casinos to make money! The "Law of Large Numbers" was proposed by the mathematician Bernoulli. It is said that n(a) is the number of occurrences of a in n independent repeated experiments, and p is the probability of occurrence of a in each experiment. When n is large enough At this time, for any positive number ε, there is lim{[|(n(a)/n)| p]<ε}=1, the formula is so complicated that 99% of gamblers can’t understand it, it doesn’t matter if you don’t understand it, we Just look at the result, the money that the banker wins in the end = 0.02*a.
The money made by the dealer is ultimately only related to the player's bet size! This is what we often call "flowing water". As long as the players keep playing, the dealer will keep making money! Regardless of whether the player loses or wins, the banker always wins! Why does the casino have a "minimum betting amount", because expanding the "turnover" can maximize profits!
So don't think how smart you are, you have to be thankful that you haven't played long enough, that's why you lose out of ten bets.
As long as you enter the gamble, you are a poor ghost.
Let's go further, even if the probability of both parties is equal, you are still a loser. This involves "infinite wealth" and "the law of gamblers losing everything". This theorem has many in real life. Applications, such as "surname demise" and "mitochondrial Eve hypothesis", under the condition of equal probability, whoever has the most capital has the highest winning rate.
You and I bet against each other, you and I each have 5 yuan, until we lose it all. Then you have a 50% chance of winning and a 50% chance of losing.
You bet against me, you have 5 yuan, I have 10 yuan, until you lose all, then the probability of your winning is only 33.3%, and the probability of losing is 66.7% (this involves Gaussian probability theory and Taylor's Series theory), hidden behind is the casino boss Kelly formula, which will be described in detail in the following subsections.
For small retail investors, casinos can generally think that wealth is infinite, you can't win it, but it can eat you. In the eyes of the casino owner, there are only two kinds of people in the world: one is poor now, and the other is poor in the future.
The "Law of Unlimited Wealth" also explains why casinos set maximum bets. It’s not that the boss is kind enough to protect gamblers from going bankrupt, it’s just that the boss has set up a safety barrier to protect himself. Imagine that one day Bill Gates goes to the casino to have fun and spend tens of billions in it at once, the casino boss will really cry Yes, although this kind of thing is unlikely to happen, but it must be guarded against, so the casino designs the highest betting amount according to its own wealth ability, that is, to resist the "infinite wealth theorem"!
Casino Big Boss Kelly Formula: Let me tell you how to bet
Kelly Formula is famous in the world of advanced gamblers, so what is Kelly Formula, let’s look at an example first:
there is a simple 2 to 1 bet, flip a coin to bet, the coin If it is heads, you will get 2 yuan. If it is tails, you will lose 1 yuan. Your total assets are 100 yuan, and you can invest any amount in each bet.
How would you bet on it?
If you are an adventurist, you may think, if you want to play, you can play the big ticket, and bet all 100 yuan at once. If you are lucky, you can get 200 yuan for a head, which is another gambling history worth showing off; but , If you lose, you have to hand over 100 yuan of assets to the other party, and you will have nothing. It is definitely not a good idea to come to Las Vegas with great difficulty.
If you're a conservative, you might think, be careful, take your time with the one percent. You only bet 1 yuan each time, you win 2 yuan on heads, and lose 1 yuan on tails. After playing 20 games, I suddenly felt that the opponent bet 10 yuan once and won 20 yuan, and I only won 2 yuan once, and 10 times to win 20 yuan. I regret that I have missed hundreds of millions!
100 is too much and 1 yuan is too little, what percentage should be invested? Ordinary gamblers seem to have no solution, but the Kelly formula tells you that the answer is 25%!
Let's take a look at the true face of Kelly's formula:
In the formula, the meanings of each parameter are:
f* = capital ratio to bet
p = probability of winning
q = probability of failure
b = odds
The numerator bp-q above the formula represents "Winning face" is called "expectation value" in mathematics.
What is the right bet for no more, no less? Kelly told us that by choosing the best betting ratio, we can obtain the highest profit in the long run. Going back to the example mentioned above, the probability of heads and tails of the coin toss is 50%, so the probability of p and q winning and losing is 0.5, and the odds = expected profit ÷ possible loss = 2 yuan profit ÷ 1 yuan loss , the odds are 2, and the answer we want is f, which is (bp - q) ÷ b = (2 * 50% - 50%) ÷ 2 = 25%.
Take out 25% of the funds to bet in order to maximize the profit of the game.
Every time a casino operator makes a bet, he will keep the mathematical principles in mind. As an ordinary gambler, apart from silently saying "Bodhisattva Blessing" in his heart, how can he know the mathematical knowledge behind it.
Therefore, even if you win the support of the God of Wealth, you will never win the "Kelly Formula".
In fact, the author of the formula, Kelly, is not a senior gambler, but a famous physicist. When he invented this formula, he was a research scientist in the famous Bell Labs. His research direction was still new at that time. Cutting-edge TV signal transmission protocol.
Except for 100% win, all casino games should not be wagered at any time
, and almost all of them are unfair games to the gambler.
But this kind of unfairness does not mean that the banker is cheating. Modern casinos rely on mathematical rules to earn profits. In a sense, casinos are the most transparent and open places. Stanley Ho was afraid that nine lives would not be enough.
The Kelly formula is not conceived out of thin air. This mathematical model has been verified on Wall Street. In addition to being regarded as a righteous god in casinos, it is also known as the "artifact of money management". It is the favorite of investment tycoons such as Bill Gross, Buffett Relying on this formula also made a lot of money.
In June 1955, an extremely famous TV program called 64000 dollar question appeared in the United States. The answerers accumulate bonuses by answering the questions continuously, and it became popular all over the United States for a while, with a ratings of 85% during prime time, and there are many counterfeit programs from all walks of life. Such a question-and-answer show quickly attracted off-market betting to bet on the winner. The show was recorded in New York, live on the East Coast, and time-lapse on the West Coast. Some scandals broke out in the news at the time. The gamblers on the West Coast learned the results in advance by phone and rushed to place bets before the West Coast live broadcast.
After Kelly watched the news, he thought that the problem of how to maximize the long-term benefits of gamblers who have some inside information but also some noise can be solved by using the formulas of their laboratory on consulting and noise transmission research. Therefore, he launched the prototype of the Kelly formula with a horse racing model.
Kelly's theory is this. For horse racers who have certain inside information, the first natural thought is of course to put in all the funds, but this will cause a miserable situation in case of losing everything. In the problem that Kelly wants to solve, losing all the funds at any one time is obviously not in line with the need to maximize cumulative returns.
What we should really care about is the long-term accumulated income. For the accumulated income, the final result is only related to the number of rounds won or lost, and has nothing to do with the order of winning or losing. So he introduced an optimal investment position ratio to maximize the long-term cumulative return:
bet = edge / odds = expected profit / profit return
edge=bp-q
where edge can be understood as the probability of winning in gambling ＊Odds - the probability of failure, that is, the odds of winning mentioned above. When the number of edge is positive, this is a game worth betting on, and when the edge is 0 or negative, it means that the gambler does not have edge and should not bet.
The odds are odds, and we can understand it as a public estimate of probability, which is public information.
We can use Kelly to simulate such a situation: Xiao Ming now has a starting capital of 100 yuan, and he will now flip a coin 4 times, and each time he flips a coin for heads, he will get 6 times the capital return (1 with 5) , when he flips a coin tails, accompany the light. May I ask how Xiaoming should allocate the funds for each bet in order to maximize his profit after 4 coin tosses?
According to the Kelly formula calculation, we can establish such a probabilities of 50%, edge = 0.5*5-0.5 = 2, odds is 5, the best position is 40%, we can see that there are 16 possible positions in the end Of the results (4 throws), 12.96 and 8100 appeared 1 time, 64.8 and 1620 appeared 4 times, 324 appeared 6 times, and the payoff of 16 results was 324. The purpose of the Kelly formula is to maximize the payoff from these outcomes.
Since the Kelly formula focuses on long-term rate of return and risk control, it naturally attracts investors to apply it in investment. For example, after the famous legendary mathematician Edward Thorp read Kelly’s thesis, he first taught himself Fortran and developed a set of algorithms for blackjack on an IBM mainframe (interested students can go to the movie 21, the card counting in the movie The way to get edge is the source of edge), and brought Kelly's mentor to Las Vegas to attract a lot of money.
Conclusion - the only rule to win:
no one can convince a depraved gambler without gambling, because this is a defect of personality.
But if you are still a rational person, stop being obsessed with so-called luck.
What gamblers can rely on is the blessing of their ancestors, and the bosses behind the casino are such great gods as Gauss, Kelly, and Bernoulli.
How could you win the dealer?
In terms of rationality, no one is more rational than a casino owner.
When it comes to mathematics, no one is more proficient in mathematics than the experts hired by casino owners.
In terms of gambling capital, no one has more capital than the casino owner.
Add
