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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:.

Games of chance are also good examples of combinations , permutations and arrangements, which are met at every step: 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.

These can be identified with elementary events that the event to be measured consists of. 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 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.

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.

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. 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.

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