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 5 to 1 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 double and 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, the average house profit will be 10 x $1 x 5.26% = $0.53. Of course, it is not possible for the casino to 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 vary greatly with the game. Keno can have house edges up to 25%, 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 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%.
The luck factor in a casino game is quantified using standard deviations (SD). The standard deviation of a simple game like Roulette can be calculated using the binomial distribution. In the binomial distribution, SD = sqrt (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. Therefore,
SD (Roulette, even-money bet) = 2b sqrt(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 per round, the standard deviation will be 2 x 1 x sqrt(10 x 18/38 x 20/38) = $3.16. After 10 rounds, the expected loss will be 10 x $1 x 5.26% = $0.53. As you can see, standard deviation is many times the magnitude of the expected loss.
The range is six times the standard deviation: three above the mean, and three below. Therefore, after 10 rounds betting $1 per round, your result will be somewhere between -$0.53 - 3 x $3.16 and -$0.53 + 3 x $3.16, i.e., between -$10.01 and $8.95. (There is still a 0.1% chance that your result will exceed a $8.95 profit, and a 0.1% chance that you will lose more than $10.01.) This demonstrates how luck can be quantified; we know that if we walk into a casino and bet $5 per round for a whole night, we are not going to walk out with $500.
The standard deviation for Pai Gow poker is the lowest out of all common casinos . Many , particularly slots, have extremely high standard deviations. As the size of the potential payouts increase, so does the standard deviation.
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, such as Mike Shackleford, the "Wizard of Odds".