The Mathematics Behind Casino Gaming

Know the odds before you go.
Marcia Wendorf
Math formulas behind cards, dice, and chips1, 2

With COVID lifting, you might be thinking of visiting Las Vegas or many of the world's other casino districts. Before you go, you should be aware of the mathematics behind many of the casino's most popular games.

The fact is that casinos don't pay out at true odds, and this difference is called the house advantage. Over time, all casino games favor the house, however, some games have a lower house advantage than others, and knowing that can make the difference between coming home flush or coming home broke.

The casino games with the lowest house advantage are blackjack, craps, baccarat, and table and video poker. If you play blackjack according to basic blackjack strategy, you can reduce the house's advantage to virtually zero. In craps, you can reduce the house's advantage if you bet the Pass Line with Odds, Come with Odds, or make a Place bet on the numbers 6 or 8.

Baccarat offers the low house advantage of 1.06 percent on the Banker's Hand and 1.24% on the Player's hand, and if you search out a 9/6 Jacks or Better video poker machine, you'll increase your odds of winning against the house.

Experiments, events, event occurrences, sets, and sample spaces

The game of craps involves throwing dice. In the roll of a die (one half of a pair of dice), the throwing of the die is an experiment, the outcome of the throw is called an event, and the total number of possibilities is called the sample space. For a single die, the sample space is the set {1, 2, 3, 4, 5, 6}.

When rolling two dice, the sample space is a set of 36 ordered pairs: {(1, 1), (1, 2), (1, 3), ... (1, 6), (2, 1), (2, 2), ... (6, 5), (6, 6). The sample space for rolling an even number with two dice is comprised of six possibilities: {(2, 2), (2, 4), (2, 6), (4, 4), (4, 6) and (6, 6)}.


Craps table
Craps table Source: Sergey Korotkih/iStockPhoto

Craps is played on a table 12 - 14 feet (4.27 - 1.83 m) long with an identical layout on each side. A maximum of 16 players can play craps at one time, eight on each side.

The dice are passed around clockwise, and when it's your turn, you simply roll the dice to the opposite end of the table. This first roll is called the come out roll. If you roll a 7 or 11, that's called a natural, and you win and get to roll again. If you roll a 2, 3, or 12, that's called craps, and you lose, but you get to roll again.

If you roll a 4, 5, 6, 8, 9, or 10, that number becomes the point, and the object then becomes to roll that number again before rolling a 7. If you roll a 7 before rolling the point, you lose, and the dice are passed to the next shooter. If you roll your point before rolling a 7, you have made a pass, and you win and get to roll again.

Craps is based on the fact that 7 is the most likely outcome of any roll of two dice because it can be formed in any of six ways:

  • Total 2 - (1, 1) = 1/36
  • Total 3 - (1, 2) (2, 1) = 2/36
  • Total 4 - (1, 3), (3, 1) (2, 2) = 3/36
  • Total 5 - (1, 4), (4, 1) (2, 3), (3, 2) = 4/36
  • Total 6 - (1, 5), (5, 1), (2, 4), (4, 2) (3, 3) = 5/36
  • Total 7 - (1, 6), (6, 1), (2, 5), (5, 2) (3, 4), (4, 3) = 6/36
  • Total 8 - (2, 6), (6, 2), (3, 5), (5, 3), (4, 4) = 5/36
  • Total 9 - (3, 6), (6, 3), (4, 5), (5, 4), = 4/36
  • Total 10 (4, 6), (6, 4), (5, 5) = 3/36
  • Total 11 - (5, 6) (6, 5) = 2/36
  • Total 12 - (6, 6) 1/36

Therefore, your odds of rolling a 4 or a 10 are 2 - 1, your odds of rolling a 5 or a 9 are 3 - 2 and your odds of rolling a 6 or an 8 are 6 - 5.


Roulette table
Roulette table Source: Hendra Su/iStockPhoto

In roulette, the experiment is the spinning of the wheel, the event is the ball falling into a numbered slot, and the sample space for American roulette is comprised of 38 possibilities: {1, 2, 3, ... 36, 0, 00}. European roulette wheels don't include the 00, so their sample space is comprised of 37 possibilities, and therefore, they have better odds than their American counterparts.

The event occurrence of a getting a red number is: {1, 3, 5, 7, 9, 12, 14, 16, 18, 19, 21, 23, 25, 27, 30, 32, 34, 36}, while the event occurrence of getting a black number is: {2, 4, 6, 8, 10, 11, 13, 15, 17, 20, 22, 24, 26, 28, 29, 31, 33, 35}.

On an electric board above the table, casinos display the numbers that have come up on the last 20 spins, and this lets you "chart the wheel." Statistically, any particular number should hit only once in 38 spins, so if particular numbers appear more often than that, the wheel might have a bias.

The roulette table is divided into two areas: inside bets and outside bets. The inside area consists of the squares containing the numbers 1 through 36 and the 0 and 00 pockets. The outside area is everything else. In the US, all Inside Bets have a 5.26 percent house advantage, with the exception of the 5-Number Bet, whose house advantage is 7.89 percent.

The best betting strategy for winning at roulette is the Column or Dozen Bet. By placing a wager on two Columns or two Dozens, it allows you to cover almost 2/3 of the numbers.


Blackjack table
Blackjack table Source: mphillips007/iStockPhoto

Blackjack is unique among casino table games in that the numerical house advantage is only 0.5 percent if you play using Basic Blackjack Strategy. For players who don't use this strategy, the house advantage rises to 2.74 percent.

Casinos play blackjack with one, two, four, six, or eight decks. The fewer decks being used, the more advantageous it is to the player because they can count cards. Blackjack is also the only casino game whose rules vary depending on the casino. For example, in Atlantic City, all casinos use six or eight decks that are dealt from a shoe, and the cards are dealt face up.

In Las Vegas, some casinos play by those rules, while others use only one or two decks, and the players' cards are dealt face down. The rules can vary even among different tables in some Las Vegas casinos.

Before sitting down at one of a blackjack table's seven seats, check for a sign on the table that displays that table's minimum and maximum bets; for example, "$5 minimum $2,000 maximum." You should also know that casinos change their bet limits depending on how busy they are.

The object of blackjack is not to beat the other six players, but to beat the dealer. You beat the dealer when:

  • Your cards total 21 or less, and your total is greater than the dealer's total
  • The dealer busts, which means his cards total more than 21.

Each card in blackjack is worth its numerical value, all face cards are worth 10, and aces are worth either 1 or 11. For example, if you're dealt an ace and a 7, your hand is worth either 8 or 18. Hands that contain an ace are called soft hands, and hands that don't contain an ace are called hard hands.

The dealer always deals themself two cards, one face up and one face down. The dealer's face down card is called his hole card. After all players have received their cards, the dealer turns over his hole card and decides whether to hit (draw more cards) or stand (keep the cards he has).

Dealers must follow strict rules: if their cards total 16 or less, they must hit. If their cards total 17 or higher, they must stand. If their cards total a soft 17, they must hit. When dealers hit on a soft 17, it's bad for players because there is a good chance that the dealer can improve his hand without busting.

Basic Blackjack Strategy

Basic blackjack strategy has been developed over the years and tested through millions of computer simulations. If you've read about blackjack wizards or students from Ivy League colleges winning millions from casinos, this is the system they were using.

Basic Blackjack Strategy is based on two things — the dealer's up card and statistics that state that the dealer will bust more often when his up card is a 2, 3, 4, 5, or 6, and will win more hands when his up card is a 7, 8, 9, 10-value card or an ace.

Counting Cards

Counting cards
Counting cards Source: AlexSava/iStockPhoto

Card counting is not illegal, but since you are a guest of the casino, they can ask you to leave if they observe you counting cards. Card counting is based on the simple premise that the more 10-value cards remaining in the deck, the better it is for the player, and the more low value cards remaining in the deck, the worse it is for the player.

To keep track of what's left in the deck or shoe, card counters keep a tally in their head of +1 for every 2, 3, 4, 5 or 6 that has been dealt, and -1 for every 10, jack, queen, king and ace that has been dealt. They ignore 7, 8 and 9. Counters must keep track of every card that's been dealt — to themselves, to other players, and the dealer.

You can vary your bets depending on the count. The higher the count, the higher your bet should be, but be wary, this is exactly the sort of behavior casinos look for in determining who is counting cards.

In single-deck games, dealers place a plastic card inside the deck to alert them that it's time to reshuffle. The deeper into the deck the plastic card is placed, its penetration point, the better for the card counter.

Slot Machines

Slot machines
Slot machines Source: andresr/iStockPhoto

Slot machines of yesteryear were mechanical, while today's slots are computers. All slot machines contain a microchip called a Random Number Generator (RNG) and each number corresponds to a particular position of the reels at the pay line.

The maximum number of reel positions for a given machine is determined by the number of reels, and the number of symbols on each reel. For example, if a machine has 3 reels and 25 symbols per reel, then the total number of combinations is:
25 x 25 x 25 = 15,625.
The RNG for this machine would cycle between 1 and 15,625, and unfortunately for players, only one of those combinations wins.

The best strategy for winning at slots is to play the maximum number of coins that a machine allows. This is because the largest jackpots are only achieved with the maximum number of coins in.

Casinos know the average rate of return on their slots, and many advertise it, as in: "The loosest slots in Las Vegas." For a casino whose average rate of return is 85 percent, you can expect to win $85 for every $100 you wager.

While slots deservedly have a reputation as "one armed bandits," you can maximize your chances of winning and keep your losses to a minimum by switching machines often, and not "chasing your losses" with a particular machine.

If the casino where you are playing slots offers a slot club, join it. Once you fill out a form at the service desk, you'll be given a card resembling a credit card. Place that card in a slot marked on the machine, and you will start earning points for every coin you play. You can then redeem those points for free rooms, meals, show tickets, and even cash, further minimizing your losses.

Video Poker

Video poker
Video poker Source: spxChrome/iStockPhoto

Like slot machines, video poker machines come in various flavors. The most common machines are Jacks or Better, but Jokers Wild, Deuces Wild, and Bonus Poker machines are also common. Jokers Wild machines add at least one Joker to the standard 52-card deck.

While slot machines require absolutely no skill to play, video poker machines require that you understand the ranking of poker hands. The order of hierarchy for poker hands is:

  • Royal flush (10, J, Q, K, A in one suit)
  • Straight flush (five consecutive cards in a single suit)
  • Four of a kind (four of the same value card in the four suits)
  • Full house (three of one value card and two of another value card)
  • Flush (five cards all in the same suit)
  • Straight (five consecutive value cards in any suit)
  • Three of a kind (three of the same value card)
  • Two pair (two pairs of the same value card)
  • Pair of jacks or better (two jacks, queens, kings or aces),

Not all Jacks or Better video poker machines are created equal, some have a lower house advantage. To find these machines, look at the pay table posted above the video screen, specifically at the payouts for one coin for a full house and a flush. Some machines pay 9 - 1 for a full house, and 6 - 1 for a flush. These are called 9/6 machines. Other video poker machines pay 8 - 1 for a full house, and 5 - 1 for a flush. Before you sit down at a video poker machine, look at their pay table, and compare the payouts for a full house and a flush.

Progressive jackpot video poker machines, which are linked with other machines in the same casino or even among machines in many casinos, offer gigantic payouts, however, some casinos lower the regular payouts of these machines to add to the progressive jackpot. Other casinos take a percentage of every wager to add to the progressive jackpot. Only play these machines if you're prepared for a possible loss, of feeling lucky, and play the maximum number of coins.

Baccarat and Mini-Baccarat

Baccarat table
Baccarat table Source: Sergey_Korotkih/iStockPhoto

Pronounced "bah-ca-rah", this game conjures up images of men in tuxedos and women in evening gowns and furs, dripping in diamonds. In some countries, baccarat is called chemin de fer.

Baccarat tables seat up to 15 players, while mini-baccarat tables seat six or seven players. In both baccarat and mini-baccarat, the player has only one choice to make — whether to bet with the "player", the dealer (also known as the banker), or on a tie. If a hand results in a tie, the dealer pays those who bet on a tie, and all bets on the player or dealer are pushes, meaning that while they are not paid, their wager is not collected.

Once players make their choice of whether to bet with the dealer or the player, the dealer deals two, 2-card hands face up from an 8-deck shoe, one to himself and one to a "player". In baccarat, the "player" hand isn't dealt to an actual player, but is dealt right next to the dealer's hand. The object of the game is for the cards to total 9 or as close to 9 as possible.

In baccarat, aces count as 1, cards count their face value, and 10s and face cards count as 0, so a hand with a Queen and a 4 would count as 4, and a hand with an ace and a 7 would count as 8. If, however, a hand totals above 9, you must subtract 10 from that hand. For example, a 9 and an 8 would count as 7 and a 5 and 5 would count as 0.

A total of either 8 or 9 on those first two cards is called a natural, with the 8 called le petit natural and the 9 called le grand natural. A 9 always beats an 8. If both hands are 8s or 9s, the game ends in a tie; if the player hand is a natural, the dealer folds. If neither hand is a natural, then the dealer must do the following, always dealing to the player hand first:

  • 0 - 5, deal a card
  • 6 or 7, stand
  • 8 or 9, stand (it's a natural).

The house advantage on the Player bet is 1.24 percent, and the payoff on the Player's hand is 1 - 1. The payoff on the Banker's hand is 1 - 1, less a 5 percent commission, so a $5 wager on the Dealer's hand would return $4.75. The house advantage on the Banker's bet is 1.06 percent. The payoff on a tie is either 9 - 1 or 8 - 1, depending on the casino, however since the true odds of a tie occurring are 10 - 1, the Tie bet is the worst bet in baccarat. The house advantage is 9.5 percent at the 9 - 1 payoff, and 18.5 percent at the 8 - 1 payoff.

There is no effective strategy to winning at baccarat, so only wager an amount that you can afford to lose.

Texas Hold-Em Poker

Texas Hold-Em
Texas Hold-Em Source: olio/iStockPhoto

As Formula 1 is to auto racing, Texas Hold 'Em is to casino gaming. Texas Hold-Em was first introduced in 1963 in Las Vegas by a group of Texas gamblers, including Doyle Brunson and Amarillo Slim. The three most common variations of Hold 'em are limit, no-limit, and pot-limit.

The most popular form in U.S. casinos is limit hold 'em, in which bets and raises during the first two betting rounds (pre-flop and flop) must be equal to the big blind (the call price of the round); this amount is called the small bet. In the next two betting rounds (turn and river), the bets and raises must be equal to twice the big blind; this amount is called the big bet.

The form of Texas Hold 'em most commonly viewed on televised poker tournaments, and played in the World Series of Poker, is no-limit. Players can bet or raise any amount over the minimum raise, including all of their chips, which is called an all-in bet.

Texas Hold 'em hands that are worth staying in for are:

  • A high pair (ten, jack, queen, king, and ace)
  • An ace and king
  • An ace and queen
  • An ace and jack
  • A king and queen
  • A king and jack
  • A queen and jack
  • Same suited jack and 10
  • Same suited king and 10
  • Same suited 10 and 9
  • A pair of nines
  • Same suited queen and 10
  • Same suited 9 and 8
  • Same suited 8 and 7.

You should play more aggressively when there are five or fewer players at your table. The fewer the players, the greater the chances that the cards you need are still in the deck. Any time five or more players see the flop, chances are good they are holding aces, kings, or queens. This means there is a better opportunity for small pairs, low straights or flush draws.

The ratio of the size of the pot compared to the amount you have to put in to the pot to stay in the hand is called the pot odds. For example, if the pot is five big bets and you have to call two big bets to stay in, then you are getting pot odds of 5 - 2.

In Texas Hold 'em, you also need to compute the number of outs. These are any unseen cards that, if drawn, will improve a player's hand to one that is likely to win. For example, a hand with four diamonds has nine outs to make a flush because there are a total of 13 cards in each suit. For a hand comprised of two small pairs, to achieve a full house there are four outs: the two remaining cards of each value.

Enjoy gambling

The best way to enjoy gambling is to set an amount of money that you want to gamble with, and are prepared to lose, and stick to that amount. At casino resorts, especially those in Las Vegas, there are plenty of things to see and do besides gambling, such as swimming, golfing, shopping, eating, free nightly performances, musical acts, and even museums.

If you take advantage of all these forms of entertainment, besides gambling, you'll return home rested, relaxed, and with a fatter wallet.

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