Introduction to Gambling
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. 1/4*12=3. Similarly, Mel also gets half of the two probabilities, i.e. (1+1/2)2=3/4, i.e. 3/4*12=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!
Experiments, events and probability spaces
The technical processes of a game stand for experiments that generate aleatory events. Here are a few examples:
Throwing the dice in craps is an experiment that generates events such as occurrences of certain numbers on the dice, obtaining a certain sum of the shown numbers, and obtaining numbers with certain properties (less than a specific number, higher than a specific number, even, uneven, and so on). The sample space of such an experiment is {1, 2, 3, 4, 5, 6} for rolling one die or {(1, 1), (1, 2), ..., (1, 6), (2, 1), (2, 2), ..., (2, 6), ..., (6, 1), (6, 2), ..., (6, 6)} for rolling two dice. The latter is a set of ordered pairs and counts 6 x 6 = 36 elements. The events can be identified with sets, namely parts of the sample space. For example, the event occurrence of an even number is represented by the following set in the experiment of rolling one die: {2, 4, 6}.
Spinning the roulette wheel is an experiment whose generated events could be the occurrence of a certain number, of a certain color, or a certain property of the numbers (low, high, even, uneven, from a certain row or column, and so on). The sample space of the experiment involving spinning the roulette wheel is the set of numbers the roulette holds: {1, 2, 3, ..., 36, 0, 00} for the American roulette, or {1, 2, 3, ..., 36, 0} for the European. The event occurrence of a red number is represented by the set {1, 3, 5, 7, 9, 12, 14, 16, 18, 19, 21, 23, 25, 27, 30, 32, 34, 36}. These are the numbers inscribed on the roulette wheel and table, in red.[3]
Dealing cards in blackjack is an experiment that generates events such as the occurrence of a certain card or value as the first card dealt, obtaining a certain total of points from the first two cards dealt, exceeding 21 points from the first three cards dealt, and so on. In card games, we encounter many types of experiments and categories of events. Each type of experiment has its own sample space. For example, the experiment of dealing the first card to the first player has as its sample space the set of all 52 cards (or 104, if played with two decks). The experiment of dealing the second card to the first player has as its sample space the set of all 52 cards (or 104), less the first card dealt. The experiment of dealing the first two cards to the first player has as its sample space a set of ordered pairs, namely all the 2-size arrangements of cards from the 52 (or 104). In a game with one player, the event the player is dealt a card of 10 points as the first dealt card is represented by the set of cards {10♠, 10♣, 10♥, 10♦, J♠, J♣, J♥, J♦, Q♠, Q♣, Q♥, Q♦, K♠, K♣, K♥, K♦}. The event the player is dealt a total of five points from the first two dealt cards is represented by the set of 2-size combinations of card values {(A, 4), (2, 3)}, which counts 4 x 4 + 4 x 4 = 32 combinations of cards (as value and symbol).
In the 6/49 lottery, the experiment of drawing six numbers from the 49 generates events such as drawing six specific numbers, drawing five numbers from six specific numbers, drawing four numbers from six specific numbers, drawing at least one number from a certain group of numbers, etc. The sample space here is the set of all 6-size combinations of numbers from the 49.
In draw poker, the experiment of dealing the initial five card hands generates events such as dealing at least one certain card to a specific player, dealing a pair to at least two players, dealing four identical symbols to at least one player, and so on. The sample space in this case is the set of all 5-card combinations from the 52 (or the deck used).
Dealing two cards to a player who has discarded two cards is another experiment whose sample space is now the set of all 2-card combinations from the 52, less the cards are seen by the observer who solves the probability problem. For example, if you are in play in the above situation and want to figure out some odds regarding your hand, the sample space you should consider is the set of all 2-card combinations from the 52, less the three cards you hold and less the two cards you discarded. This sample space counts the 2-size combinations from 47.
The probability model
A probability model starts from an experiment and a mathematical structure attached to that experiment, namely the space (field) of events. The event is the main unit probability theory works on. In gambling, there are many categories of events, all of which can be textually predefined. In the previous examples of gambling experiments, we saw some of the events that experiments generate. They are a minute part of all possible events, which is the set of all parts of the sample space.
For a specific game, the various types of events can be:
Events related to your play or opponents’ play;
Events related to one person's play or several persons’ play;
Immediate events or long-shot events.
Each category can be further divided into several other subcategories, depending on the game referred to. These events can be defined, but they must be done very carefully when framing a probability problem. From a mathematical point of view, the events are nothing more than subsets and the space of events is a Boolean algebra. Among these events, we find elementary and compound events, exclusive and nonexclusive events, and independent and non-independent events.
In the experiment of rolling a die:
Event {3, 5} (whose literal definition is the occurrence of 3 or 5) is compound because {3, 5}= {3} U {5};
Events {1}, {2}, {3}, {4}, {5}, {6} are elementary;
Events {3, 5} and {4} are incompatible or exclusive because their intersection is empty; that is, they cannot occur simultaneously;
Events {1, 2, 5} and {2, 5} are nonexclusive, because their intersection is not empty;
In the experiment of rolling two dice one after another, the events obtaining 3 on the first die and obtaining 5 on the second die are independent because the occurrence of the second event is not influenced by the occurrence of the first, and vice versa.
- In the experiment of dealing the pocket cards in Texas Hold’em Poker:
- The event of dealing (3♣, 3♦) to a player is an elementary event;
- The event of dealing two 3's to a player is compound because is the union of events (3♣, 3♠), (3♣, 3♥), (3♣, 3♦), (3♠, 3♥), (3♠, 3♦) and (3♥, 3♦);
- The events player 1 is dealt a pair of kings and player 2 is dealt a pair of kings are nonexclusive (they can both occur);
- The events player 1 is dealt two connectors of hearts higher than J and player 2 is dealt two connectors of hearts higher than J are exclusive (only one can occur);
- The events player 1 is dealt (7, K) and player 2 is dealt (4, Q) are non-independent (the occurrence of the second depends on the occurrence of the first, while the same deck is in use).
These are a few examples of gambling events, whose properties of compoundness, exclusiveness and independency are easily observable. These properties are very important in practical probability calculus.
The complete mathematical model is given by the probability field attached to the experiment, which is the triple sample space—field of events—probability function. For any game of chance, the probability model is of the simplest type—the sample space is finite, the space of events is the set of parts of the sample space, implicitly finite, too, and the probability function is given by the definition of probability on a finite space of events.
Combinations
Games of chance are also good examples of combinations, permutations, and arrangements, which are met at every step: combinations of cards in a player's hand, on the table, or expected in any card game; combinations of numbers when rolling several dice once; combinations of numbers in lottery and bingo; combinations of symbols in slots; permutations and arrangements in a race to be bet on and the like. Combinatorial calculus is an important part of gambling probability applications. In games of chance, most of the gambling probability calculus in which we use the classical definition of probability reverts to counting combinations. The gaming events can be identified with sets, which often are sets of combinations. Thus, we can identify an event with a combination.
For example, in a five draw poker game, the event at least one player holds a four of a kind formation can be identified with the set of all combinations of (xxxxy) type, where x and y are distinct values of cards. This set has 13C(4,4)(52-4)=624 combinations. Possible combinations are (3♠ 3♣ 3♥ 3♦ J♣) or (7♠ 7♣ 7♥ 7♦ 2♣). These can be identified with elementary events that the event to be measured consists of.
The mathematical principles in the casino
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, the 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, but 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, adhering to the principle of a positive rate of return, do not gamble for a long time and will lose the gambling game, only to gamble on a sure win. They are 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, 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.
Expectation and strategy
Games of chance are not merely pure applications of probability calculus and gaming situations are not just isolated events whose numerical probability is well established through mathematical methods; they are also games whose progress is influenced by human action. In gambling, the human element has a striking character. The player is not only interested in the mathematical probability of the various gaming events, but he or she has expectations from the games while a major interaction exists. To obtain favorable results from this interaction, gamblers take into account all possible information, including statistics, to build gaming strategies. The oldest and most common betting system is the martingale, or doubling-up, system on even-money bets, in which bets are doubled progressively after each loss until a win occurs. This system probably dates back to the invention of the roulette wheel. Two other well-known systems, also based on even-money bets, are the d’Alembert system (based on theorems of the French mathematician Jean Le Rond d’Alembert), in which the player increases his bets by one unit after each loss but decreases it by one unit after each win, and the Labouchere system (devised by the British politician Henry Du Pré Labouchere, although the basis for it was invented by the 18th-century French philosopher Marie-Jean-Antoine-Nicolas de Caritat, marquis de Condorcet), in which the player increases or decreases his bets according to a certain combination of numbers chosen in advance. The predicted average gain or loss is called expectation or expected value and is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical odds are repeated many times. A game or situation in which the expected value for the player is zero (no net gain nor loss) is called a fair game. The attribute fair refers not to the technical process of the game, but to the chance balance house (bank)–player.
Even though the randomness inherent in games of chance would seem to ensure their fairness (at least with respect to the players around a table—shuffling a deck or spinning a wheel do not favor any player except if they are fraudulent), gamblers always search and wait for irregularities in this randomness that will allow them to win. It has been mathematically proved that, in ideal conditions of randomness, and with negative expectation, no long-run regular winning is possible for players of games of chance. Most gamblers accept this premise, but still work on strategies to make them win either in the short term or over the long run.
House advantage or edge
Casino games provide a predictable long-term advantage to the casino, or "house" while offering the player the possibility of a large short-term payout. Some casino games have a skill element, where the player makes decisions; such games are called "random with a tactical element." While it is possible through skillful play to minimize the house advantage, a player rarely has sufficient skill to eliminate his inherent long-term disadvantage (the house edge or house vigorish) in a casino game. The common belief is that such a skill set would involve years of training, extraordinary memory, and numeracy, and/or acute visual or even aural observation, as in the case of wheel clocking in Roulette. For more examples see Advantage Gambling.
The player's disadvantage is a result of the casino not paying winning wagers according to the game's "true odds", which are the payouts that would be expected considering the odds of a wager either winning or losing. For example, if a game is played by wagering on the number that would result from the roll of one die, true odds would be 5 times the amount wagered since there is a 1/6 probability of any single number appearing. However, the casino may only pay 4 times the amount wagered for a winning wager.
The house edge (HE) or vigorish is defined as the casino profit expressed as a percentage of the player's original bet. In games such as Blackjack or Spanish 21, the final bet may be several times the original bet, if the player doubles or splits.
Example: In American Roulette, there are two zeroes and 36 non-zero numbers (18 red and 18 black). If a player bets $1 on red, his chance of winning $1 is therefore 18/38 and his chance of losing $1 (or winning -$1) is 20/38.
The player's expected value, EV = (18/38 x 1) + (20/38 x -1) = 18/38 - 20/38 = -2/38 = -5.26%. Therefore, the house edge is 5.26%. After 10 rounds, play $1 per round, and the average house profit will be 10 x $1 x 5.26% = $0.53. Of course, the casino can't win exactly 53 cents; this figure is the average casino profit from each player if it had millions of players each betting 10 rounds at $1 per round.
The house edge of casino games varies greatly with the game. Keno can have house edges up to 25% and slot machines can have up to 15%, while most Australian Pontoon games have house edges between 0.3% and 0.4%.
The calculation of the Roulette house edge was a trivial exercise; for other games, this is not usually the case. Combinatorial analysis and/or computer simulation are necessary to complete the task.
In games that have a skill element, such as Blackjack or Spanish 21, the house edge is defined as the house advantage from optimal play (without the use of advanced techniques such as card counting or shuffle tracking), on the first hand of the shoe (the container that holds the cards). The set of the optimal plays for all possible hands is known as "basic strategy" and is highly dependent on the specific rules, and even the number of decks used. Good Blackjack and Spanish 21 games have to house edges below 0.5%.
Online slot games often have a published return to player (RTP) percentage that determines the theoretical house edge. Some software developers choose to publish the RTP of their slot games while others do not. Despite the set-theoretical RTP, almost any outcome is possible in the short term.
Standard deviation
The luck factor in a casino game is quantified using standard deviation (SD). The standard deviation of a simple game like Roulette can be simply calculated because of the binomial distribution of successes (assuming a result of 1 unit for a win, and 0 units for a loss). For the binomial distribution, SD is equal to , where is the number of rounds played, is the probability of winning, and is the probability of losing. Furthermore, if we flat bet at 10 units per round instead of 1 unit, the range of possible outcomes increases 10 fold. Therefore, SD for Roulette even-money bet is equal to , where is the flat bet per round, is the number of rounds, , and .
After enough large number of rounds the theoretical distribution of the total win converges to the normal distribution, giving a good possibility to forecast the possible win or loss. For example, after 100 rounds at $1 per round, the standard deviation of the win (equally of the loss) will be . After 100 rounds, the expected loss will be .
The 3 sigma range is six times the standard deviation: three above the mean, and three below. Therefore, after 100 rounds betting $1 per round, the result will very probably be somewhere between and , i.e., between -$34 and $24. There is still a ca. 1 to 400 chance that the result will be not in this range, i.e. either the win will exceed $24, or the loss will exceed $34.
The standard deviation for the even-money Roulette bet is one of the lowest out of all casinos games. Most games, particularly slots, have extremely high standard deviations. As the size of the potential payouts increase, so does the standard deviation.
Unfortunately, the above considerations for small numbers of rounds are incorrect, because the distribution is far from normal. Moreover, the results of more volatile games usually converge to the normal distribution much more slowly, therefore much more huge number of rounds are required for that.
As the number of rounds increases, eventually, the expected loss will exceed the standard deviation, many times over. From the formula, we can see the standard deviation is proportional to the square root of the number of rounds played, while the expected loss is proportional to the number of rounds played. As the number of rounds increases, the expected loss increases at a much faster rate. This is why it is practically impossible for a gambler to win in the long term (if they don't have an edge). It is the high ratio of short-term standard deviation to expected loss that fools gamblers into thinking that they can win.
The volatility index (VI) is defined as the standard deviation for one round, betting one unit. Therefore, the VI for the even-money American Roulette bet is .
The variance is defined as the square of the VI. Therefore, the variance of the even-money American Roulette bet is ca. 0.249, which is extremely low for a casino game. The variance for Blackjack is ca. 1.2, which is still low compared to the variances of electronic gaming machines (EGMs).
Additionally, the term of the volatility index based on some confidence intervals are used. Usually, it is based on the 90% confidence interval. The volatility index for the 90% confidence interval is ca. 1.645 times as the "usual" volatility index that relates to the ca. 68.27% confidence interval.
It is important for a casino to know both the house edge and volatility index for all of their games. The house edge tells them what kind of profit they will make as percentage of turnover, and the volatility index tells them how much they need in the way of cash reserves. The mathematicians and computer programmers that do this kind of work are called gaming mathematicians and gaming analysts. Casinos do not have in-house expertise in this field, so they outsource their requirements to experts in the gaming analysis field.
Bingo probability
The probability of winning a game of Bingo (ignoring simultaneous winners, making wins mutually exclusive) may be calculated as:
since winning and losing are mutually exclusive. The probability of losing is the same as the probability of another player winning (for now assuming each player has only one Bingo card). With players taking part: with players and our player being designated . This is also stated (for mutually exclusive events) as .
If the probability of winning for each player is equal (as would be expected in a fair game of chance), then and thus and therefore . Simplifying yields
For the case where more than one card is bought, each card can be seen as being equivalent to the above players, having an equal chance of winning. where is the number of cards in the game and is the card we are interested in.
A player () holding cards therefore will be the winner if any of this cards win (still ignoring simultaneous wins):
A simple way for a player to increase his odds of winning is therefore to buy more cards in a game (increase ).
Simultaneous wins may occur in certain game types (such as online bingo, where the winner is determined automatically, rather than by shouting "Bingo" for example), with the winnings being split between all simultaneous winners. The probability of our card, , winning when there is either one or more simultaneous winners is expressed by:
where is the probability of there being simultaneous winner (a function of the game type and number of players) and being the (fair) probability that is one of the winning cards. The overall expected value for the payout (1 representing the full winning pot) is therefore:
Since, for a normal bingo game, which is played until there is a winner, the probability of there being a winning card, either or or ... or , and these being mutually exclusive, it can be stated that
and therefore that
The expected outcome of the game is therefor not changed by simultaneous winners, as long as the pot is split evenly between all simultaneous winners. This has been confirmed numerically.
To investigate whether it is better to play multiple cards in a single game or to play multiple games, the probability of winning is calculated for each scenario, where cards are bought.
where n is the number of players (assuming each opposing player only plays one card). The probability of losing any single game, where only a single card is played, is expressed as:
The probability of losing games is expressed as:
The probability of winning at least one game out of games is the same as the probability of not losing all games:
When , these values are equal:
but it has been shown that for . The advantage of grows both as grows and decreases. It is therefore always better to play multiple games rather than multiple cards in a single game, although the advantage diminishes when there are more players in the game.
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 zero, zero, or zero passes, you lose.
If zeros and zeros did not exist, the roulette system would be fair and perfect.
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 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 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, if 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 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 or Bentley on the road, that the money I will spend will be far greater than paying the insurance book. And the insurance pays out in their favor the insurance company.
Although there is an element of gambling, in most cases it is 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 in 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 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, 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 result is only a loss of money, this is a gambler's psychology.
Gamblers gamble 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, it's 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 for 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, and 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, 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, 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 higher than one in two, while the chances of a bottom horse winning are lower than one in two hundred.
_-_The_Cardsharps_-_Google_Art_Project.jpg)
