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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 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.

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.

For more examples see Advantage gambling. However, the casino may only pay 4 times the amount wagered for a winning wager.

In games such as Blackjack or Spanish 21 , the final bet may be several times the original bet, if the player doubles or splits.

In American Roulette , there are two zeroes and 36 non-zero numbers 18 red and 18 black. Therefore, the house edge is 5.

The house edge of casino games vary greatly with the game. The calculation of the Roulette house edge was a trivial exercise; for other games, this is not usually the case.

In games which 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 house edges below 0. 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. 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.

Furthermore, if we flat bet at 10 units per round instead of 1 unit, the range of possible outcomes increases 10 fold.

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.

The 3 sigma range is six times the standard deviation: There is still a ca. 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. 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 variance of the even-money American Roulette bet is ca.

The variance for Blackjack is ca. Additionally, the term of the volatility index based on some confidence intervals are used. It is important for a casino to know both the house edge and volatility index for all of their games.

At first sight, it looks pretty random:. A roulette wheel has the same number of red and black numbers. If we ignore the green 0 which means the casino wins we would expect the number of red and black numbers to be.

This looks pretty evenly distributed — there is a small difference between the number of red and black results, but that is always to be expected in probability.

He realised that if the results were completely random, then each of the four possible pairs of two consecutive colours should also be equally likely.

Again we can count the number of occurrences in our example:. Of course, we might have just been unlucky in this particular sequence of results — but Pearson tested many thousands of results and always found the same.

It gets even worse if we look at triples of results. Each of the 8 possible triples of colours should be equally likely, but that is clearly not the case here:.

It seems that in this particular casino, the colours alternate much more often than one would expect. Pearson calculated that the probability of seeing results which were this skewed was less than 1 in ,,!

He assumed that the Roulette wheels were rigged to create higher profits for the Casino — and wrote many angry letters to expose this scam.

When he finally travelled to Monte Carlo, he discovered that the reason for the skewed results was of a very different nature: This story shows that we humans tend to be quite bad at coming up with random-looking data: This can be used effectively to detect fraud in banking and insurance.

Here you can try for yourself if you are better than the journalists: While Pearson only analysed previous Roulette results, others tried to use mathematics to increase their chances of winning in casinos.

One of these was Edward Thorp , who invented card counting — a technique that allowed him to beat casinos at Blackjack. He later turned his focus to Roulette: After the dealer sets the roulette wheel spinning, there are just a few seconds when you are still allowed to place new bets.

Unfortunately this time is much too short for humans to calculate the outcome in their head. At the Massachusetts Institute of Technology, Thorp discussed his ideas with Claude Shannon , another mathematician and the father of information theory.

The 3 sigma range is six times the standard deviation: There is still a ca. 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. 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 variance of the even-money American Roulette bet is ca.

The variance for Blackjack is ca. Additionally, the term of the volatility index based on some confidence intervals are used. 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.

From Wikipedia, the free encyclopedia. This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources.

Unsourced material may be challenged and removed. Gambling mathematics Mathematics of bookmaking Poker probability. Casino game Game of chance Game of skill List of bets Problem gambling.

Category Commons Wiktionary WikiProject. Retrieved from " https: Applied probability Gambling mathematics Gambling. Which means sticking to the smaller mobile screen.

When Android smartphones and iPhone casinos hit the scene, Probability jumped at the chance to show off their years of experience.

They created casino games for mobile that simply work. They kept with their two original theories of what players want, especially when it comes to slot machines.

Well theory one make sense in theory, but we generally disagree. The second theory about paytables is a little more complex, but this one we can see the logic.

In land-based or online casinos you have more time to play. You sit, you chill, you take your time to get that jackpot. So the paytable reflects that, giving you wins more frequently in a small space of time.

On one hand you can play all their games on tablet. On the other, their casino and platform is definitely tailored for the smaller screen.

Their potential winning depends on the likelihood of each of these outcomes. Here is one of the many hundreds of newspaper extracts which Pearson collected and analysed.

At first sight, it looks pretty random:. A roulette wheel has the same number of red and black numbers. If we ignore the green 0 which means the casino wins we would expect the number of red and black numbers to be.

This looks pretty evenly distributed — there is a small difference between the number of red and black results, but that is always to be expected in probability.

He realised that if the results were completely random, then each of the four possible pairs of two consecutive colours should also be equally likely.

Again we can count the number of occurrences in our example:. Of course, we might have just been unlucky in this particular sequence of results — but Pearson tested many thousands of results and always found the same.

It gets even worse if we look at triples of results. Each of the 8 possible triples of colours should be equally likely, but that is clearly not the case here:.

It seems that in this particular casino, the colours alternate much more often than one would expect. Pearson calculated that the probability of seeing results which were this skewed was less than 1 in ,,!

He assumed that the Roulette wheels were rigged to create higher profits for the Casino — and wrote many angry letters to expose this scam.

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