Why the casino will always win

Casinos exist to make profits, they do so from patrons who place bets on various games that are offered. Yet many casino patrons are relatively uninformed about the games they are playing.

When you go to a cinema, you know how much you pay for your entertainment. When you go to a casino, you do not know how much you will win or lose - it will depend on the spin of a wheel, the turn of a card.

But you should expect to lose, and you should know how much to expect to lose, because this information is available through a quantity known as the house margin or house edge.

Every bet made at the casino has a house margin. Suppose the house margin for a particular bet is 2 per cent. This means that if you place this bet many times, you will lose about 2 per cent of the total amount of money you wagered.

For a more concrete example, suppose a friend offers you the following $1 bet on the toss of a coin. If the coin lands on heads you lose your $1. If the coin lands on tails you get back your original $1 wager and receive $0.50 profit. Assuming the coin is fair, so that both heads and tails are equally likely, the house margin for this bet is 25 per cent. If you play this game 100 times, your total wager is $100, and you can anticipate losing about $25. The actual amount you lose may be greater or less than $25 depending on how the coin falls.

Suppose now that your friend offers you a different bet: You still lose if the coin lands heads, but if the coin lands on tails you now get a profit of $0.98. The house margin for this bet is 1 per cent. If you play this game 100 times, you can anticipate losing about $1, though again the actual amount you lose may be greater or less than $1. If you are lucky you may see a profit, but this becomes less likely the more you play.

The two bets outlined lead to the observation that, in percentage terms, you can anticipate paying less money at the casino if you place bets with lower house margins. Lower house margins are to your advantage; they will cost you less because you can anticipate a higher proportion of your total wager being returned. For the same level of entertainment, a lower house margin provides a better deal for patrons.

Let's consider the house margins for roulette, Sic Bo (Tai Sai), baccarat, craps and blackjack.


Roulette

THE roulette wheels typically used in Europe and Australia have 37 slots or pockets. A ball is spun into the wheel and comes to rest in one of the slots. Bets can be placed based on the outcome of the spin. We assume here that the roulette wheel is fair and the ball is equally likely to land in any of the 37 slots.

The wheel has 18 red slots and 18 black slots. The remaining slot, denoted by the numeral 0, is typically coloured green. Consider betting $1 on the outcome that the ball lands in a red slot. If this occurs, your bet wins, and the casino pays you even money, which means that you get $2; your $1 wager and $1 profit. On the other hand, if the ball lands in a black or green slot, you lose your $1 wager. The green slot gives the casino the advantage, because the ball will more likely land in one of the 19 black/green slots than in one of the 18 red slots. This gives the casino a house margin of 2.70 per cent. If you make this bet 100 times, you can expect the cost of playing to be $2.70, though the actual amount that you lose may be greater or less than $2.70.

The money that the casino pays out on winning roulette bets is calculated so that whatever bet you make, the house margin is always 2.70 per cent. If you spend a long time at the roulette table, expect it to cost you 2.70 per cent of the total amount of money you wager. An American roulette wheel has an additional slot denoted by 00. This additional slot favours the casino; the house margin for American roulette is 5.26 per cent.


Sic Bo

Sic Bo is a dice game which is also known by the names 'Tai Sai' or 'Big and Small'. Different bets can be placed based on the values obtained by three dice. Unlike roulette, different bets give different house margins. The bet known as 'Big' wins if the sum of the three dice is 11 or greater, with the exception that the bet loses if the three dice all show the same number. The exception gives the casino the advantage. For a winning bet on 'Big' the casino pays you even money. Without the exception, losing or winning the bet would be equally likely. With the exception, a loss is more likely than a win. The house margin for the bet 'Big' is 2.78 per cent.

The bet known as 'Small' is a bet on the sum of the three dice being 10 or less, again with the exception that the bet loses if the three dice all show the same number. The house margin is again 2.78 per cent.

Another possibility is to place a bet on all three dice being equal to six. On average this will happen only once in every 216 occasions. For a $1 wager, a winning bet on this very unlikely event means that the casino gives you $180 profit, resulting in an unfavourable house margin of 16.20 per cent. Some casinos give you only $150 profit, which results in a hugely unfavourable house margin of 30.09 per cent.

The house margins of 2.78 and 16.20 per cent are typically the lowest and highest house margins of all bets that can be made at the game of Sic Bo. This means that if you spend a long time playing Sic Bo, expect it to cost you between 2.78 and 16.20 per cent of the total amount of money you wager, with the precise percentage depending on the type of bets that you make.


Baccarat

Baccarat is a card game where the 'player' plays against the 'banker'. Neither the player nor banker makes any decision that influences the outcome of the game, which is completely determined by the order of the cards.

There are three outcomes: a player win, a banker win or a tie. A version of the game known as mini-baccarat has the same rules, payouts and house margins. The only difference is that mini-baccarat is typically faster, and therefore more bets are made in any given period.

You may bet on the banker to win the card game. If the player then wins the card game, you lose the bet. If there is a tie the wager is returned to you. Similarly, if you make a bet on the player, the bet loses if the banker wins, and the wager is returned if there is a tie. A $1 winning bet on the player pays even money and therefore gives you a $1 profit, whereas a $1 winning bet on the banker typically gives you a $0.95 profit. Usually the game is played with eight decks of cards, in which case the house margins immediately after a card shuffle are 1.06 per cent for betting on the banker and 1.24 per cent for betting on the player. You may also be able to bet on a tie, but this bet is unfavourable and is best avoided.

In a series of games the house margins may be slightly different, but these values are an accurate approximation, assuming that you do not keep track of the cards that have been dealt in previous rounds of play.

A skilled player may use this information to decrease the house margins, but only by small amounts.

Some casinos also offer a variation of the game where a winning $1 bet on the banker pays a $1 profit unless the banker's winning hand has a value of six, in which case you get a $0.50 profit. Under this variation, the house margin for betting on the banker increases to 1.46 per cent.


Craps

Craps is a dice game where each round consists of rolling two dice one or more times. The dice are rolled by casino patrons, and the value of the roll is the sum of the two dice. There are numerous possible bets that can be placed, and they have different house margins. I will consider one popular bet which is synonymous with the game of craps.

The first roll in each round is known as the come out roll. If the come out roll is 7 or 11, the bet wins. If the come out roll is 2, 3 or 12, the bet loses. If any other number is rolled, this number is known as the point. The two dice are then rolled until either the point or a seven is rolled; in the former case the bet wins, while in the latter case the bet loses. A winning bet pays even money. The casino again gets the advantage because a loss is more likely than a win. The house margin for this bet is 1.41 per cent.

Assumptions
A question that is often asked is whether the house margins for these games can be altered. The calculation of the house margins are based on mathematical results that cannot be challenged.

However, the calculations are also based on assumptions about the game and these assumptions can be challenged. For example, in Sic Bo, one assumption is that all three dice are fair, so that each of the six numbers on each die are equally likely to appear. If the dice were unfair, then the house margin would not be as given here. It seems reasonable to assume that the dice in Sic Bo are fair.

Suppose that you play one of the four games described above and you accept that the assumptions on which these house margins are based are reasonable. The assumptions include fair die rolls, well-shuffled cards and fair roulette wheels. These house margins are then fixed. Whatever choices you make, you cannot change these house margins and you cannot make any decisions which will affect them.

However, there are casino games for which you can affect the house margins. One of the most popular is blackjack.

Blackjack is very different to any of the games above, because you have to make decisions during the game, and the decisions you make can affect whether you win or lose. To analyse such games, we need to know in advance what the player intends to do. Suppose that before the game begins, the player decides on exactly what decisions he or she will make under any given circumstances. This set of decisions is known as the player strategy. Once the player strategy is known, the house margin can be approximated.

Assuming that the player does not keep track of the cards that have been dealt in previous rounds, the strategy that achieves the minimum house margin is known as the Basic Strategy. The game of blackjack has many different variations in the rules implemented by different casinos, and the Basic Strategy depends on the exact rules that are implemented. Moreover, the different rules mean that different house margins can be obtained at different casinos.

For most casinos, the house margins that can be obtained under the Basic Strategy are between 0.4 and 0.6 per cent. But these house margins can only be obtained if the player makes no errors in playing the Basic Strategy. In practice, errors may be made and the house margins will then be larger.

While the house margin gives you valuable information about the bet you are making, it does not tell you everything. For example, placing 100 $1 bets on red at a roulette table is not the same as placing 10 $10 bets on red, and yet in both cases we have the same house margin, at 2.70 per cent, and the total wager is the same.

For both of these scenarios, you expect to lose 2.70 per cent of $100, or $2.70, though the amount you actually lose may be greater or less than $2.70.

The difference lies in the statistical concept of variation. For 100 $1 bets, the amount you actually lose will typically be closer to $2.70 than for 10 $10 bets because there is less variation, and therefore less risk involved.

The house margins tell you how much money you can expect to lose.

It is easy to forget that casino house margins are percentages, and this has an important consequence: The more money you gamble, the more money you can expect to lose.

Alec Stephenson
for The Straits Times Interactive, 17 October 2005

The writer is an assistant professor of statistics at the National University of Singapore.