The games available at most casinos are usually called casino games. In casino games, players wager cash or casino chips on a variety of possible random outcomes or combinations of outcomes. Casino games are also available at online casinos, where permitted by law. Casino games may also be played outside of casinos for entertainment purposes such as at parties or school competitions, some on machines that simulate gambling.

Categories

There are three general categories of casino games: gaming machines, table games, and random number games. Gaming machines, such as slot machines and pachinko machines, are usually played by one player at a time and do not require the involvement of casino employees to play. Table games, such as blackjack or craps, involve one or more players competing against the casino’s own dealer rather than each other. Table games are usually run by a casino employee known as a croupier or dealer. Random number games are based on the selection of random numbers, either from a computerized random number generator or from other gaming equipment. Random number games can be played at a table or through the purchase of paper tickets or cards, such as keno or bingo.

Some casino games combine several of the above; for example, Roulette is a table game run by a dealer, which involves random numbers. Casinos may also offer other types of games, such as hosting Poker games or tournaments, in which players compete against each other.

House edge

Casino games typically provide a predictable long-term advantage to the casino, or “house”, while offering players the possibility of short-term gains that can in some cases be large. Some casino games have an element of skill, where the player’s decisions impact the outcome. Players who have sufficient skill to eliminate the inherent long-term disadvantage (house edge or power) in a casino game are referred to as advantage players.

The players’ disadvantage is the result of the casino not paying out winning bets according to the game’s “true odds,” which are the expected payouts given the odds of the bet either winning or losing. For example, if a game is played by betting on the number that will be rolled on a single die, the true odds are 5 times the amount wagered because there is a 1 in 6 chance of any given number coming up, assuming the player gets his original wager back. However, the casino can only pay out 4 times the amount wagered on a winning bet.

The house edge or power is defined as the casino’s advantage expressed as a percentage of the player’s original bet. (In games like blackjack or Spanish 21, the final bet may be several times the original bet, if the player doubles and splits.)

In American roulette, there are two “zeros” (0, 00) and 36 non-zero numbers (18 red and 18 black). This leads to a higher house edge compared to European roulette. The probability of a player, betting 1 unit on red, winning is 18/38 and the probability 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 advantage is 10 × 1 × 5.26% = 0.53 units. The European roulette wheel has only one “zero” and therefore the house edge (ignoring the prison rule) is 1/37 = 2.7%.

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

Calculating the house edge in roulette is a trivial exercise; for other games, this is usually not the case. Combinatorial analysis and/or computer simulations are required to complete the task.

In games that have an element of skill, such as blackjack or Spanish 21, the house edge is defined as the house advantage from optimal play (without using sophisticated techniques such as card counting), on the first hand of the shoe (the container that holds the cards). The set of optimal plays for all possible hands is known as “basic strategy” and depends heavily on the specific rules and even the number of decks used.

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

In games where the player is not competing against the house, such as poker, the casino usually makes money through a commission, known as the “rake”.

Standard Deviation

The luck factor in casino games is quantified using the standard deviation (SD).[2] The standard deviation of a simple game such as roulette can be calculated using the binomial distribution. In the binomial distribution, SD = npq, where n = number of spins played, p = probability of winning, and q = probability of losing. The binomial distribution assumes an outcome of 1 unit for a win, and 0 units for a loss, rather than 1 unit for a loss, which doubles the range of possible outcomes. Next, if we bet flat at 10 units per spin instead of 1 unit, the range of possible outcomes increases 10-fold.[3]

SD (roulette, even money bets) = 2b npq, where b = flat bet per spin, n = number of spins, p = 18/38, and q = 20/38.

For example, after 10 spins at 1 unit per spin, the standard deviation is 2 × 1 × 10 × 18/38 × 20/38 = 3.16 units. After 10 spins, the expected loss is 10 × 1 × 5.26% = 0.53. As you can see, the standard deviation is many times the expected loss.[4]

The standard deviation for pai gow poker is the lowest of all common casino games. Many casino games, especially slot machines, have very high standard deviations. The larger the potential payout size, the more the standard deviation can increase.

As the number of spins increases, eventually the expected loss will exceed the standard deviation, many times over. From the formula, we can see that the standard deviation is proportional to the square root of the number of spins played, while the expected loss is proportional to the number of spins played. As the number of spins increases, the expected loss increases at a much faster rate. This is why it is impossible for a gambler to win in the long run. It is the high ratio of short-term standard deviation to expected loss that fools gamblers into thinking they can win.

It is important for casinos to know the house edge and variance for all of their games. The house edge tells them what kind of profit they will make as a percentage of their turnover, and the variance tells them how much they need as a cash reserve. The mathematicians and computer programmers who do this kind of work are called game mathematicians and game analysts. Casinos do not have in-house expertise in this area, so they outsource their needs to experts in the field of game analysis.

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