Casino Advantage, Table games

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Roulette Mathematical house advantage

In the early frontier gambling saloons, the house would set the odds on roulette tables at 27 for 1. This meant that on a $1 bet you would get $27 and the house would keep your initial dollar.

Today most casino odds are set by law, and they have to be either 34 to 1 or 35 to 1. This means that the house pays you $34 or $35 and you get to keep your original $1 bet.

The house average or house edge (also called the expected value) is the amount the player loses relative to any bet made, on average. If a player bets on a single number in the American game there is a probability of 1/38 that the player wins 35 times the bet, and a 37/38 chance that the player loses their bet. The expected value is:
−1×37/38 + 35×1/38 = −0.0526 (5.26% house edge)
For European roulette, a single number wins 1/37 and loses 36/37:
−1×36/37 + 35×1/37 = −0.0270 (2.70% house edge)
The presence of the green squares on the roulette wheel and on the table are technically the only house edge. Outside bets will always lose when a single or double zero come up. However, the house also has an edge on inside bets because the pay outs are always set at 35 to 1 when you mathematically have a 37 to 1 chance at winning a straight bet on a single number. To demonstrate the house edge on inside bets, imagine placing straight $1 wagers on all inside numbers on a roulette table (including 0 and 00) to assure a win. You would only get back 36 times your original bet having spent $38. The only exception are the five numbers bet where the house edge is considerably higher (7.89% on an American wheel), and the 'even money' bets in some European games where the house edge is halved because only half the stake is lost when a zero comes up.
The house edge should not be confused with the hold. The hold is the average percentage of the money originally brought to the table that the player loses before he leaves - the actual "win" amount for the casino. The Casino Control Commission in Atlantic City releases a monthly report showing the win/hold amounts for each casino. The average win/hold for double zero wheels is between 21-30%, significantly more than the 5.26% house edge. This reflects the fact that the player is 'churning' the same money over and over again. A 26.3% hold, for example, would imply that on average, the player bets the total he brought to the table five times, as 26.3% is approximately equal to 100% - (100% - 5.26%)^5. For example, a player with $100 making $10 bets on red (which has a near 50/50 chance of winning) is highly unlikely to lose all his money after only 10 bets, and will most likely continue to bet until he has lost all of his money or decides to leave. A player making $10 bets on a single number (with only 1/38 chance of success) with a $100 bankroll is far more likely to lose all of his money after only 10 bets. Despite being more likely to lose, the casino's average hold from this type of player would be significantly lower than the even-money bettor, because the single number player will on average bet less money (at 5.26% expected loss per dollar bet).






Rule variations and the "house advantage for Blackjack"
The blackjack player will encounter many rule variations which affect the house advantage and therefore affect his chances of winning. Some rules are determined by law or regulation, others by the casino itself. Not all rules are posted, so the player may have to ask either beforehand or when the situation occurs. Over 100 variations exist.
The casino has a "house advantage" at blackjack just as it does at any other casino game. The primary house advantage in blackjack comes from the fact that if the player busts he loses, irrespective of whether the dealer subsequently busts. If a particular casino game has a house advantage of 5%, it means that - over the long run - the casino will win about 5% of any initial bet. As long as the blackjack player uses the best possible strategy (a strategy which is known as "basic strategy"), the house advantage in blackjack is usually less than 1%. This is very favourable to the player compared to other casino games. Of course, many blackjack players do not know basic strategy or do not follow it, so the true house advantage in those cases may be much higher.
Dealer hits soft 17
Each casino has a rule about whether or not the dealer hits soft 17, a rule which is printed on the table itself. In the "S17" game, the dealer stands on all 17s. In the "H17" game, the dealer hits on soft 17s. Of course, the dealer always stands on hard 17s. In either case, the dealer has no choice; he either must or must not hit. The "Hit soft 17" game is less favourable to the player with about a 0.2% higher house advantage.
Number of decks
The number of decks used has a major effect on the player's chance of winning, because it affects the house advantage. All things being equal, fewer decks are always more favourable for the basic strategy player. One cause of this is that player blackjack is slightly more likely in single deck blackjack (because blackjack requires two different cards, by removing a card of one type (e.g., a Ten), getting one of a different type (e.g., an Ace) is more likely - and the effect is much greater in a single deck game than in a multi-deck game), and if the player does have blackjack, the dealer is significantly less likely to have blackjack as well (which is a push), meaning that statistically the player should get paid at 3:2 more often in the single deck game.
When single deck blackjack is offered, it is generally offered with more restrictive rules that favour the house. For illustrative purposes, the statistics below all use the same rules: double after split, re split to four hands, one card to split Aces, no surrender, double on any two cards, original bets only lost on dealer blackjack, dealer hits soft 17, and cut-card used. The single deck game is much better than double deck, which is significantly better than four decks, while from six decks and up there is very little difference.
Number of Decks
House Advantage
Single deck
0.17%
Double deck
0.46%
Four decks
0.60%
Six decks
0.64%
Eight decks
0.66%
Surrender
Some casinos offer a favourable option called "surrender," which allows the player to give up half his bet and not play out the hand. This option is sometimes referred to as "late" surrender because it occurs after the dealer has checked his or her hole card for a blackjack. When casinos first opened in Atlantic City, the surrender option was available before the dealer checked for blackjack - a rule highly advantageous to the player - but this "early surrender" option soon disappeared. Early surrender variations still exist in several countries.
The player should only surrender on the very worst hands, because having even a 25% chance of winning will result in a better average return than giving up half of his bet. With early surrender, a player is more likely to surrender against a dealer Ace.
Resplitting
If the player splits a pair other than aces and a third card of that value appears, the player can usually split again (or "re split") by putting up another bet equal to the original bet. Then there will be three bets on the table and three separate hands. Some casinos allow unlimited re-splitting of cards other than aces, while others may limit it to a certain number of hands, such as four hands (for example, "re split to 4").
Hit/re split split aces
After splitting aces, one common rule is that only one card will be dealt to each ace; the player cannot split, double, or take another hit on either hand. Rule variants include allowing re-splitting aces or allowing the player to hit split aces. Allowing the player to hit aces reduces the casino edge by about 0.13%, allowing re-splitting aces reduces the edge by about 0.03%. While re-splitting aces is not uncommon, houses that allow the player to hit split aces are extremely rare.
Double after split
After splitting a pair, most casinos allow the player to "double down" on either or both of the new two-card hands. This is called "double after split" and provides an advantage to the player of about 0.12%.
Double on 9/10/11 or 10/11 only
Often called "Reno" rules, this rule restricts the player to doubling down only on an initial player total of 10 or 11 (sometimes 9, 10, or 11 - more common in Europe). It prevents doubling on soft hands such as soft 17 (ace-6), and is unfavourable for the player. It increases the house advantage by between 0.09% (8 decks) and 0.15% (1 deck) for the 9-11 rule, and between 0.17% (8 decks) and 0.26% (single deck) for the 10-11 rule. These numbers can vary due to interaction with other rules.
No hole-card
In most non-U.S. casinos, a 'no hole card' game is played. This means that there is no dealer hole card. This usually affects the player's strategy when deciding whether to double and/or split, since a dealer blackjack will result in the loss of the split and double bets. For instance, holding 11 against a dealer 10, the correct strategy is to double in a hole card game (where the player knows the dealer's second card is not an ace), but to hit in a no hole card game. The no hole card rule adds approximately 0.11% to the house edge.
In some places,[where?] if the dealer is later found to have blackjack, the player loses only his original bet but not any additional bets (doubles or splits). This has the same advantage as the hole-card game, and as such does not change basic strategy.
Altered payout for blackjack
In many casinos in Nevada--usually at tables with the lowest table minimums--a blackjack hand pays only 6:5 or even 1:1 instead of the usual 3:2. This is the most unfavourable common variation, increasing the house edge significantly more than most U.S. player restrictions. Since blackjack occurs in approximately 4.8% of hands, the 1:1 game increases the house edge by 2.3%, while the 6:5 game adds 1.4% to the house edge. The 1:1 payout for video blackjack is a key reason why it has never approached the table version in terms of popularity. The 6:5 rule is most commonly employed on table blackjack at single deck games - which are otherwise the most attractive game for a basic strategy player.
Dealer wins ties
Allowing the dealer to win all push hands is catastrophic to the player. Though rarely used in standard blackjack, it is sometimes seen in "blackjack-like" games such as in some charity casinos.
Insurance
If the dealer's upcard is an ace, the player is offered the option of taking insurance before the dealer checks his or her 'hole card'.
Insurance is a side bet of up to half the original bet placed on a special portion of the table usually marked "Insurance Pays 2 to 1". This side bet is offered only when the dealer's exposed card is an ace. The idea is that the dealer's second card has a fairly high probability (nearly one-third) to be ten-valued, giving the dealer a blackjack and almost certainly results in a loss for the player. It is attractive (although not necessarily wise) for the player to insure against this possibility by making an "insurance" bet, which pays 2-to-1 if the dealer has a blackjack, in which case the "insurance proceeds" will make up for the concomitant loss on the original bet. The insurance bet is lost if the dealer does not have blackjack, although the player can still win or lose on the original bet.
Insurance is a poor bet for the player unless he is counting cards because, in an infinite deck, 4/13 of the cards have a value of ten (10, J, Q, or K) and 9/13 therefore are not, so the theoretical return for an infinite deck game is 4/13 * 2 * bet - 9/13 * bet = -1 /13 * bet, or -7.69%. In practice, the average house edge will be lower than this, because by eliminating even one non-ten card from the shoe (the dealer's ace), the proportion of the remaining cards that are valued at ten is higher. Even so, the bet is generally to be avoided, as the house's average edge is still more than 7%.
A player who is counting cards can keep count of the remaining tens in the shoe and use it to make insurance bets only when he has an edge (e.g., when more than one third of the remaining cards are tens). In addition, in a multi-hand single deck game, it is possible for insurance to be a good bet simply by observing the other cards on the table - for an initial hand, if the dealer has an ace, then there are 51 cards left in the deck, of which 16 are tens. However, if there are as few as 2 players playing, and none of their two initial cards are tens, then that means that 16 out of 47 remaining cards are ten - better than 1 in 3, and so the insurance bet is a good one.
When the player has blackjack and the dealer has an ace, the insurance bet may be offered as "even money", meaning that the player's blackjack is paid immediately at 1:1 before checking the dealer's hand. 'Even money' is only a slightly different bet; the difference being that the player must have sufficient funds to insure the blackjack if even money is not offered. Taking even money is generally even worse than average, because one of the player's two cards is a ten, so the proportion of tens remaining in the deck is lower.
In casinos where a hole card is dealt, a dealer who is showing a card with a value of ace or 10 may slide the corner of his hole card over a small mirror or electronic sensor on the tabletop in order to check whether he has a blackjack. This practice minimizes the risk of inadvertently revealing the hole card, which may give the sharp-eyed player a considerable advantage.
Side bets
Some casinos offer a side bet with their blackjack games. Examples include side bets based on getting three 7s, a three card poker-style bet, a pair, and many others. For the side bet, the player will typically put up an additional wager alongside his main bet and can win or lose the side bet, regardless of the main game result. The house edge for side bets is usually much higher than the main game.



House Advantages of Three Card Poker
House advantages vary according to the pay scale used:
Pair Plus
United Kingdom

Straight flush
35 to 1
40 to 1
40 to 1
Three of a kind
33 to 1
30 to 1
30 to 1
Straight
6 to 1
6 to 1
6 to 1
Flush
4 to 1
4 to 1
3 to 1
Pair
1 to 1
1 to 1
1 to 1
House Advantage
2.70%
2.32%
7.28%
Ante and Play house advantage assumes the player plays optimum strategy against the dealer which is to make the Play wager with a Queen-6-4 or better. There is less pay scale variation on the Ante Bonus than on the Pair Plus with following being by far the most common.
Ante Bonus
 
Straight Flush
5 x 1
Three of a kind
4 x 1
Straight
1 x 1

Ante and Play
 
House advantage
2.01% as a percentage of total Ante and Play wager
House advantage
3.37% as a percentage of Ante wager only
There are two ways of expressing the against dealer hand house advantage. If the house advantage is expressed as a percentage of the Ante wager only (at 3.37%) this could encourage players to erroneously think that the house advantage on the Play wager is 0.00%, and so think that the Play wager should always be made. Consequently it is better for players to understand the concept of house advantage on the total combined wager in the against dealer mode, being 2.01%.
Strategy for Three Card Poker
Optimum player strategy against the dealer is to make the Play wager with Queen-6-4 or better. However the simple strategy of Queen high or better produces a very similar result. As with all negative expectation gambling games there is no legitimate strategy that can overcome the house advantage. Therefore the player should play the strategy that provides the most enjoyment. The most common wager strategies are to play Ante and Pair Plus wagers for equal amounts, or for one of the Ante and Pair Plus wagers to be double the amount of the other one of the Ante and Pair Plus wagers.
Variation of Three Card Poker
Prime Table Games has added a wager called Prime in United Kingdom casinos and the game is known as Prime Three Card Poker. The Prime wager is optionally placed before cards are dealt and pays on the colour of the player cards. If all three cards are the same colour the payoff is 3 to 1. However when included with the dealer hand if all six cards are the same colour then the payoff is increased to 4 to 1. The house advantage on this wager is 3.62%.
Online Three Card Poker
Prime Table Games has licensed Three Card Poker use by certain internet casinos that do not access United States players, though a software provider, Wager Works. These internet casinos are identifiable by reference to Prime Table Games on these sites.
The majority of internet use of Three Card Poker, including all United States use, has not been licensed. Certain internet casinos have used other game names, but the game is still the same. Players are not informed by these internet casinos that Three Card Poker is a proprietary game and that these internet casinos use the game without permission of and without payment to the game inventor or the game rights owners.
It is the opinion of Prime Table Games that this use without permission is commercial theft of intellectual property and deception of players by internet casinos and associated parties. Further, it demonstrates a lack of adequate regulation of an immature internet gambling industry.