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Casino and gambling can be a lot of fun, especially when players do not understand the mathematics behind it, but being aware of these matters would offer players more fundamental understanding of the way probabilities and odds work.

When players say they know how to beat slot machines at a casino, it really means increasing their chances of winning at slots. As usual, you’ll be relying largely on pure chance. Dec 09, 2016 If you want to know the probability of the ball landing in a green pocket, it’s 2/38, or 1/19, or 18 to 1. If you want to know the probability of the ball landing in a black pocket, it’s 18/38, or 9/19, or 37 to 18. The probability of the ball landing in the red pocket is the same.

Odds and probabilities are some of the most important aspects of gambling mathematics. They could be related even to the pay-off of a certain game. For example, there is an even money situation when it is explained that a bet pays off at the same odds as the probability of winning the bet.

Probabilities

Probabilities are often described as the very backbone of gambling mathematics.

Informally, probability is understood as the chance or the likelihood that a certain event will happen. It is also considered to be an estimate of the corresponding average frequency that is applied for a certain event to occur in a series of independent trials that have been repeated.

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The relative frequency of an event is somewhere between the probability of the event never to occur or always to occur, which basically means that it is always between 0% and 100%. Knowing how to use the probability would provide casino players with the opportunity to predict the eventual occurrence of an event. Unfortunately, it would not give them the chance to predict the exact moment when the event is to occur.

In addition, probability in gambling mathematics can be used when determining the conditions for acquiring certain results or even some financial expectations of a game in particular in the long term. Probabilities could also be helpful when determining how favourable a certain game would be and is it worth playing. This aspect of the matter is usually expressed as odds, as well as a fraction or a decimal fraction. Still, casino players should be aware of the fact that odds and probabilities are different things.

The estimate of the likelihood of a player winning divided by the total amount of available chances is called probability. The latter is also an ordinary fraction that can be expressed as a proportion between 0 and 1, or as a percentage.

What casino players should understand first is the formal theory of probability and are recommended to start with the term that is known as the “sample space”. It may sound somehow strange, but as a matter of fact, this is purely a description of all outcomes possible, or other words – of everything that can possibly happen. It is quite normal for sample spaces to be very large when it comes to casino games. As a matter of fact, they simply reflect the idea that too many things could actually happen to count them all.

The sample space is something that may occur in a large number of experiments and can normally be understood easily when considering the nature of the experiment. Still, the sample space does not have to be described in a explicit way.

Players will need be aware of some details in order to calculate the probability of a certain event. First, they will need a full count of the number of individual elements in the sample space. In addition, they will need to know the number of the individual elements related to the event. Once a player knows these values, they will be able to define the probability of the event by using the equation:

P(event) = Size of event / Size of the sample space

As shown in the equation, the letter P is used as a substitute of the word “probability”. What is not so evident by the above-mentioned equation is how the player should count the size of a variety of collections. Unfortunately, the counting problems in the gambling industry can be too complex. Of course, probabilities can be quickly calculated when counting is easy.

Odds

The most general definition of odds is the probability of a certain event happening in comparison to the probability of the same event not happening.

Odds are considered as ratios of the player's chances of losing to the likelihood of winning. In other words, they are the average frequency of a player's eventual loss to the average frequency of a win. As already mentioned above, odds are often mistaken with probabilities, but they are not the same thing.

As a matter of fact, odds almost never mean the actual likelihood of generating a win when it comes to gambling. In most cases, the term is used as a reference to a subjective estimate of the odds, and not as an exact mathematical calculation.

For example, if a player owns 1 of a total of 4 tickets, their probability to win is 1 in 4. Their odds, on the other hand, are 3 to 1. In order for a player to convert odds to probability, they need to take the likelihood for them to win, then use it as the numerator and divide by the total number of chances, including the ones for both winning and losing.

For example, let us take that the odds are 4 to 1. Then, the probability would be found by using the following equation:

1 / (1+4) = 1/5

In other words, the probability in this case would equal to 1/5, or 20%. What players need to know is the fact that the so-called “evens” are odds of 1 to 1. They are always paid even money.

The term “odds” could also be used as a substitute of the true odds that are the actual chances of winning. There are also payout odds, which represent the ratio of payout for each unit bet.

Just like probabilities, odds can also be expressed as a ratio of two numbers. The first number comes to represent the expected frequency of the occurrence of a specific outcome, and this is actually exactly what probabilities are. The second number, however, represents only the number of the rest of the possible outcomes. In comparison to probabilities, which could be expressed as percentages, odds are always displayed as ratios.

Games available in most casinos are commonly called casino games. In a casino game, the players gamble cash or casino chips on various possible random outcomes or combinations of outcomes. Casino games are also available in online casinos, where permitted by law. Casino games can also be played outside casinos for entertainment purposes like in parties or in school competitions, some on machines that simulate gambling.

Categories[edit]

There are three general categories of casino games: gaming machines, table games, and random number games. Gaming machines, such as slot machines and pachinko, are usually played by one player at a time and do not require the involvement of casino employees to play. Tables games, such as blackjack or craps, involve one or more players who are competing against the house (the casino itself) rather than each other. Table games are usually conducted by casino employees known as croupiers or dealers. Random number games are based upon the selection of random numbers, either from a computerized random number generator or from other gaming equipment. Random number games may be played at a table or through the purchase of paper tickets or cards, such as keno or bingo.

Some casino games combine multiple of the above aspects; for example, roulette is a table game conducted by a dealer, which involves random numbers. Casinos may also offer other type of gaming, such as hosting poker games or tournaments, where players compete against each other.

Common casino games[edit]

Notable games that are commonly found at casinos include:

House advantage[edit]

Casino games typically provide a predictable long-term advantage to the casino, or 'house', while offering the players the possibility of a short-term gain that in some cases can be large. Some casino games have a skill element, where the players' decisions have an impact on the results. Players possessing sufficient skills to eliminate the inherent long-term disadvantage (the house edge or vigorish) in a casino game are referred to as advantage players.

The players' 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 in 6 chance of any single number appearing, assuming that the player gets the original amount wagered back. However, the casino may only pay 4 times the amount wagered for a winning wager.

The house edge or vigorish is defined as the casino profit expressed as the 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 double and splits.)

In American roulette, there are two 'zeroes' (0, 00) and 36 non-zero numbers (18 red and 18 black). This leads to a higher house edge compared to European roulette. The chances of a player, who bets 1 unit on red, winning is 18/38 and his chances of losing 1 unit is 20/38. The player's expected value is EV = (18/38 × 1) + (20/38 × (−1)) = 18/38 − 20/38 = −2/38 = −5.26%. Therefore, the house edge is 5.26%. After 10 spins, betting 1 unit per spin, the average house profit will be 10 × 1 × 5.26% = 0.53 units. European roulette wheels have only one 'zero' and therefore the house advantage (ignoring the en prison rule) is equal to 1/37 = 2.7%.

The house edge of casino games varies greatly with the game, with some games having an edge as low as 0.3%. Keno can have house edges up to 25%, slot machines having up to 15%.

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 is necessary to complete the task.

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), 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 0.5%.

Traditionally, the majority of casinos have refused to reveal the house edge information for their slots games and due to the unknown number of symbols and weightings of the reels, in most cases it is much more difficult to calculate the house edge than that in other casino games. However, due to some online properties revealing this information and some independent research conducted by Michael Shackleford in the offline sector, this pattern is slowly changing.^[1]

In games where players are not competing against the house, such as poker, the casino usually earns money via a commission, known as a 'rake'.

Standard deviation[edit]

The luck factor in a casino game is quantified using standard deviations (SD).^[2] The standard deviation of a simple game like roulette can be calculated using the binomial distribution. In the binomial distribution, SD = √npq, where n = number of rounds played, p = probability of winning, and q = probability of losing. The binomial distribution assumes a result of 1 unit for a win, and 0 units for a loss, rather than −1 units for a loss, which doubles the range of possible outcomes. Furthermore, if we flat bet at 10 units per round instead of 1 unit, the range of possible outcomes increases 10 fold.^[3]

SD (roulette, even-money bet) = 2b√npq, where b = flat bet per round, n = number of rounds, p = 18/38, and q = 20/38.

For example, after 10 rounds at 1 unit per round, the standard deviation will be 2 × 1 × √10 × 18/38 × 20/38 = 3.16 units. After 10 rounds, the expected loss will be 10 × 1 × 5.26% = 0.53. As you can see, standard deviation is many times the magnitude of the expected loss.^[4]

The standard deviation for pai gow poker is the lowest out of all common casino games. Many casino games, particularly slot machines, have extremely high standard deviations. The bigger size of the potential payouts, the more the standard deviation may increase.

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 impossible for a gambler to win in the long term. It is the high ratio of short-term standard deviation to expected loss that fools gamblers into thinking that they can win.

It is important for a casino to know both the house edge and variance for all of their games. The house edge tells them what kind of profit they will make as percentage of turnover, and the variance 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 outsource their requirements to experts in the gaming analysis field.

See also[edit]

References[edit]

1. ^'Michael Shackleford is the wizard of odds'. Observer. Retrieved 13 October 2015.
2. ^Hagan, general editor, Julian Harris, Harris (2012). Gaming law : jurisdictional comparisons (1st ed.). London: European Lawyer Reference Series/Thomson Reuters. ISBN978-0414024861.
3. ^Gao, J.Z.; Fong, D.; Liu, X. (April 2011). 'Mathematical analyses of casino rebate systems for VIP gambling'. International Gambling Studies. 11 (1): 93–106. doi:10.1080/14459795.2011.552575. S2CID144540412.
4. ^Andrew, Siegel (2011). Practical Business Statistics. Academic Press. ISBN978-0123877178. Retrieved 13 October 2015.

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