User:Yyyyoung777/Gambling mathematics

Plan:
- Create an introduction part to gambling, its history, and its general operation to people.

- Build a part of the strategy to elaborate the analysis of statistics in gambling.

- Create a part in the end to summarize the strategy applied into a real application.

- Rewrite the language to make it less instructional for readers to read.

- References add

Introduction
In 1951, the gamblers Mel and Paul bet money. They each took out 6 gold coins to play dice beforehand and agreed that whoever won 3 games first would get 12 gold coins. The gambling was interrupted by an incident in which none of them won 3 games, at which point Paul had won one game and Mel had won two games. So, they discussed how the 12 gold coins should be divided. Paul thought that, according to the number of games they won, he should get 1/3 of the total, i.e. 4 gold coins, and Mel should get 2/3 of the total, i.e. 8 gold coins. But Mel thinks that this is unfair to him, he thinks that if the gambling continues, he is more likely to win than Paul, and he should get all 12 coins. Since the two of them could not reach an agreement, they asked Pascal and Fermat, the mathematicians of the time, for advice. This incident aroused the interest of mathematicians such as Pascal and Fermat, and they began to study how the 12 gold coins should be distributed.

Pascal thought: if another game may Paul win, may Mel win, if Paul wins, then two people win two games each, should get half of the total gold coins (recorded as 1/2); if Mel wins, then he won 3 games, can get all the gold coins (recorded as 1), Paul is not allowed to gold coins (recorded as 0). Since the probability of winning this game is equal, Paul should get half of the two probabilities, i.e. (0+1/2)2=1/4, i.e. 1/412=3. Similarly, Mel also gets half of the two probabilities, i.e. (1+1/2)2=3/4, i.e. 3/412=9.

According to Fermat: If you play two games, it is completely decisive, and its two games will have four results: (Paul wins, Paul wins), (Paul wins, Mel wins), (Mel wins, Paul wins), and (Mel wins, Mel wins). Only the first result will result in a final victory for Paul, and the other 3 results will result in a final victory for Mel. Therefore, Paul has 1/4 of the total number of gold coins, i.e. 3 coins, and Mel has 3/4 of the total number of gold coins, i.e. 9 coins.

The two mathematicians unanimously decided that Paul would get 3 gold coins and Mel would get 9 gold coins.

After this, mathematicians then became interested in gambling problems, collected a series of problems in gambling, and did further research, which eventually resulted in the first monograph in the history of the development of probability theory, "Calculation in Gambling", whose publication was one of the marks of the emergence of probability theory. From its creation in the 1750s to the present, probability theory has gone through classical probability, analytical probability, and modern probability. After nearly 340 years of development, probability theory is now inseparable from our lives.

Probability theory is a subdiscipline of mathematics with the concept of "probability" as its core. Probability is the numerical value of the probability of a chance event. Practice shows that chance events do not have a pattern in the test, but after a large number of tests, they will show some regularity. Probability theory is the mathematics of studying the laws of a large number of chance events. Since chance events exist objectively, through research, probability theory has slowly penetrated various fields and is widely used in natural science, economics, medicine, finance and insurance, and even humanities. Probability theory - the mathematics arising from gambling!

Why gamblers lose
A gambler asked Pascal why he always loses, and Pascal replied, "Because you spend too much time at the table". It is an old Chinese saying that "you will lose if you gamble for a long time"; the gambling king, Stanley Ho, also advised the world that "not to gamble is to win".

"Long time gambling will lose" reflects a basic theorem in probability theory - the law of large numbers.

When random events occur a large number of times, chance will cancel each other out, so that the arithmetic mean of the results of these events is very close to its mathematical term value in a probabilistic sense. For example, when a coin is tossed, it is random which side of the coin faces up when it falls, but when it happens enough times, the number of times the coin goes up on both sides is about one-half each.

Winning and losing gambling also behaves as a random event in a single person and for a short period of time, but if in the long run, as long as the gambler has a negative rate of return, then losing is going to happen sooner or later as the game progresses. For the casino, as long as the win rate of the gambling play is positive, it is a sure win.

The Principle of Positive Rate of Return
The key to determining victory or defeat is the rate of return as determined by the gambling rules and strategy. The rate of return reflects the truth and nature of gambling. The principle of designing gambling rules is usually to make the casino win rate slightly more than 50%, which is reflected in a positive rate of return that is slightly greater than zero. Gambling is not luck, but a contest of intellect, strategy and yield. The ultimate win of long-term gambling depends on the gambler's rate of return: if the rate of return is positive, the expected return is greater than zero and you can win; if the rate of return is negative, the expected return is less than zero and you cannot win. When the negative rate of return, "long gambling will lose" the role of the law of large numbers will increasingly appear. Professional gamblers, adhere to the principle of positive rate of return, do not gamble for a long time will lose the gambling game, only to gamble on a sure win. They are actually non-gamblers.

Law of small numbers bias
The law of large numbers means that when the sample is close to the overall, its probability will be close to the overall probability. The "law of small numbers bias" refers to the fact that the probability distribution of an event in a small sample is considered to be the overall distribution, thus exaggerating the representativeness of the small sample to the overall population. Another situation is the so-called "gambler's fallacy". For example, when flipping a coin, if it comes up heads 10 times in a row, one would think that the next time it comes up tails is very likely; in fact, the probability of coming up heads or tails is 0.5 each time, and it has nothing to do with how many times it has come up heads.

Probability is an examination of the likelihood of a phenomenon occurring in the aggregate, and cannot account for the likelihood of an individual occurrence. Ignoring the effect of sample size, believing that small and large samples have the same expected value, and replacing the correct probabilistic law of large numbers with the false psychological law of small numbers, is the cause of the great increase in people's gambling mentality. Casinos believe in the law of large numbers, and gamblers unconsciously apply the law of small numbers. The law of large numbers allows casinos to make money, and the law of small numbers allows gamblers to give money to casinos, and this is the logic of casinos' existence.

Casino Advantage
The casino advantage is the advantage that the casino has over the gamblers for each type of gambling game in the casino.

Take the coin toss for example, the chances of heads and tails are equal, 50% each, if you bet $10 on the coin landing heads up, you win, the casino pays you $10, you lose, all $10 lost to the casino, in this case, the casino advantage is zero (the casino is certainly not stupid enough to open this game); but if you win, the casino only pays you $9, you lose, but all $10 lost to the casino. The difference between winning and losing this 1 dollar, that is, the casino advantage, in the above case, the casino advantage is 10%.

In any kind of game in a casino, the casino has a certain advantage over the gamblers, and only in this way can the casino ensure that it will continue to open in the long run. The casino advantage varies greatly from game to game, with some games having a low casino advantage and others having a high casino advantage. People who gamble a lot try not to play games with a high casino advantage.

Translated with www.DeepL.com/Translator (free version)

Negative Warning of Gambling in Life:
According to conventional economic rules, casinos as places of business should not exist, that is because conventional economic rules assume that humans are rational and would predict such an outcome according to conventional economic rules.

If someone makes the simplest deal with you and you give him a thousand dollars and he gives you nine hundred and forty-seven point four dollars, then I am sure that you with a normal IQ will not accept the deal.

But strangely enough, you will see too many very smart people who will accept such a deal under certain conditions.

The vast majority of land-based casinos use the American roulette wheel, so let's take the American roulette wheel as an example. The American roulette wheel has thirty-eight numbers, one through thirty-six, plus zeros and zeroes.

Zero and French roulette communication, French roulette only a zero, its specific rules we will not speak, you only need to know one of the rules, that is, if you bet in the red box, and it happens to

If you place a bet in the red box and it happens to land on a red number, you will win the game by doubling your chips back to 100.

If you land on a black number, or are killed by a zero, zero, or zero pass, you lose.

If zeros and zeros did not exist, the roulette system would actually be a fair and perfect system.

In terms of probability, when the bet is near infinity, the percentage of winning or losing is a balanced fifty percent, but just because of the two zeros, the probability of doubling your chips is forty-seven.37 percent, meaning that every time you bet a thousand dollars, you have the potential to lose fifty-two.6 dollars.

This is the model of the deal where you just gave the other side a thousand dollars and the other side gave you nine hundred and forty-seven point four dollars. It is this tiny gap between the fair odds and the casino odds, and the other offered by the gambling establishment that allows them to make trillions of dollars a year worldwide.

As well as the outskirts of various tournaments, these shadowy betting institutions below the surface are even more frantic to eat away at the mood of every gambler.

So why would such a not so equal deal still be so appealing?

Let's continue with the trading model to consider another question, would you rather have a 100 percent probability of getting fifty dollars with certainty, or would you rather have an eighty percent chance of getting sixty-two.5?

If it were me, I would choose to prefer to have the certainty of fifty dollars.

The researchers did the survey and indeed eighty percent of the respondents chose the former. If any of you chose the latter, congratulations, you are a born gambler.

But in reality the probability values of the two options are exactly the same.

So in theory, people are not supposed to have any preference between these two options, so why is there such a choice?

Because human nature has a greater negative effect on losing fifty dollars than on winning fifty dollars positively, and the second choice brings the chance of loss, which is a negative experience.

So the value of the choice will be lower overall, a mindset we call the hate-loss principle.

The same principle is not only reflected in casinos, it is also the reason why insurance works in the insurance industry.

If gambling shouldn't exist according to traditional economic rules, then the insurance industry shouldn't exist either, because insurance is essentially the exact same nature as gambling, only in the opposite role.

In other words, the insurance industry is essentially role-swapped.

The gambling industry is just that the insurance company is the gambling and you are the casino.

Take the commercial insurance of the car for example, for example, I need to pay six thousand dollars a year for car insurance, I would never feel that the six thousand dollars spent is not worth it, instead I would feel that in this insurance deal, the six thousand dollars become worth more, because I would worry that once my car is damaged or accidentally kissed the rice rice or Bentley on the road, that the money I will spend will be far greater than paying the insurance book. And actually the insurance pays out in their favor for the insurance company.

Although there is an element of gambling, in most cases it is definitely worth it.

Overall, this creates a transaction that both the policyholder and the insurance company feel is worthwhile.

In another interesting experiment, one hundred and fifty teachers in Chicago were divided into three groups. The first group did not receive any information, while the second group was told that they would receive a scholarship at the end of their training based on their students' test scores.

The third group also had the same scholarship as the second group, the only difference was that the bonus was paid to the teachers in advance, but they were told that if the students did not achieve the required test scores, then the bonus would be refunded.

So guess what the results were?

The results of the first and second groups were almost identical, while the third group, which had been given the bonus in advance, tripled their test scores, again showing that the fear of loss is far more powerful than the promise of gain, or as I said earlier, the hatred of loss.

Fear of loss is precisely one of the major reasons why gamblers lose money.

You will find an interesting fact that a person who wins money, or picks up a sum of money, or a windfall, when the money enters his pocket, the person's subconscious mind fully assumes that he already belongs to his property.

Similarly, if a person is extremely lucky, gambling won a lot of money when, and good luck dozed off, as long as the beginning of the continuous loss of so little, he will feel that the money belonging to their own was lost, I want to get back, and never think I have won so much, losing a little is nothing.

So at this point in time there are very few people who can stop.

Of course he may also be lucky enough to win some of this money back again.

Then I can tell you that this is where the real horror begins.

Every time he gambles in the future, his mind will deeply remember and magnify this experience, and his subconscious will always tell him that the lost money can be won back again.

So this time.

He will recklessly increase the stakes, the end result is only a loss of money, this is a gambler's psychology.

Gamblers gambling nine losses, but the heart of the most profound is always the time to win money, the thrill of winning money devilishly drives them to stand next to the table again and again.

Of course not every person next to the gambling table is born, then why are there so many people gambling?

Let's go back to that wonderful odds test, still one steady gets fifty dollars, while the other becomes a twenty-five percent chance of getting two hundred dollars, twenty-five percent is getting fifty dollars, and the options are still worth the same.

But with this change in odds, respondents no longer have a preference for either, and the results of the option which is up to almost fifty percent each.

Then when we continue to change the odds, something interesting happens, do you want a 100 percent chance of getting fifty dollars or a 0.5 percent chance of getting ten thousand dollars, of course its still the same value of the option, but the odds have begun to stir up the deepest greed and toxicity of the human psyche.

Only thirty-six percent of people accept fifty dollars, while already sixty-four percent of people are willing to accept this zero point five percent of ten thousand dollars.

From this we can understand that human beings like low risk probability, but want a small chance to win more than several times or tens of times the value, not to mention hundreds of times or even more.

Probability theory is a mathematical sub-discipline formed by the concept of probability as its core.

Probability is the numerical value of the probability of the occurrence of a chance event.

The most common lottery tickets in mainland China, the lotto and the double-color ball, have a jackpot of 10 million or 5 million RMB, and the probability of the jackpot is 1 in 21.42 million and 1 in 17.72 million respectively. The probability of having three generations of grandchildren on the same day is about 1 in 270,000.

The probability of a black and white couple giving birth to black and white twins is about one in a million, and the probability of having all-male or all-female quadruplets is about one in three million and a half.

That according to the meteorological department statistics ah, the probability of a person being struck by lightning is about one in one million eight hundred thousand.

In contrast, we can find that being struck by lightning ten times may not win the jackpot once, but there will still be a very large proportion of people to buy this very small probability of lottery tickets, rather than to save money to the bank or do financial management, and thus get a small return of 100% profit.

The answer is very simple, people prefer very low probability events, the same amount of money, the interest generated by the bank will not make any difference to life.

And even if the probability of winning the lottery is extremely low, but once the prize brings a return that can change his life, which is also used to explain why humans are afraid of plane crashes, some people are therefore never afraid to fly, although the chance of a plane crash is very small, but the probability of causing death is almost 100 percent.

We can also see the end of the track.

Take horse racing for example, the best yards may have twice the odds, betting one hundred dollars to recover three hundred dollars, netting two hundred dollars.

But the bottom of the horse it put a hundred dollars can even win thirty thousand dollars, that is, two hundred times the odds, which is what we often call a blowout, cold odds are always very high, and therefore popular with gamblers.

But it turns out that, on average, the chances of a top horse winning are actually higher than one in two, while the chances of a bottom horse winning are lower than one in two hundred.