Gambling relies on numerical calculations and probabilities. Best casinos not on GamStop use mathematics to set the odds and secure a profit over time. In many games, the odds are not in the player's favour. The study of probability can help one understand the underlying risks.

Mathematical Principles

Every game in a casino begins with a simple experiment. A die roll or a roulette spin has a set of possible outcomes. These are described by a few key terms:

• Experiment – an action that produces an outcome.
  
• Event – the result of the experiment.
  
• Sample Space – the complete set of possible outcomes.
  

For instance, a single die has a sample space of {1, 2, 3, 4, 5, 6}. When two dice are thrown, the sample space consists of 36 ordered pairs. The probability of an event is the ratio of the number of favourable outcomes to the total number of outcomes.

Below is a table showing the outcomes when two dice are rolled:

Such probabilities form the basis of many casino games.

Popular Casino Games and Their Probabilities

Craps

Craps is a dice game played on a long table. Up to 16 players may participate, with eight on each side. A player rolls the dice in what is known as the come-out roll. A roll of 7 or 11 wins immediately, while a roll of 2, 3 or 12 loses. If any other number appears, it becomes the "point". The shooter must then roll the point again before a 7 appears.

The game is built on simple probability. For example, the number 7 can be rolled in six different ways, making it the most common outcome. The odds of rolling certain numbers affect the payout structure, with the chance of obtaining a 4 or 10 being lower than that of a 7.

Roulette

Roulette involves a spinning wheel and a small ball. In American roulette, the wheel has 38 pockets, including 0 and 00. European roulette removes the 00, leaving 37 pockets. This slight difference improves the player's chances on a European wheel.

Players can bet on a single number, red or black, or other groups of numbers. The layout of the betting area is divided into two sections: one for the numerical bets and another for bets on colours and odd or even outcomes. A list of features for roulette is given below:

• American Roulette: 38 pockets.
  
• European Roulette: 37 pockets.
  
• Betting on red or black covers almost half the possibilities.
  

Blackjack

Blackjack is distinct because the player competes against the dealer rather than against other players. The aim is to have a hand total closer to 21 without exceeding it. Each card from 2 to 10 has a face value; face cards are worth 10 and an ace can count as either 1 or 11.

When basic strategy is followed, the house advantage can be very low. However, if a player ignores these principles, the edge can increase significantly. Many casinos offer the game in variations with one, two, four, six, or even eight decks. Some players use card counting to track the balance of high and low cards remaining in the deck. A brief list of basic rules is shown below:

• Cards 2 to 10 hold their numerical values.
  
• Face cards are valued at 10.
  
• An ace may be counted as 1 or 11.
  
• The dealer must hit on 16 or less and stand on 17 or more.
  

Slot Machines

Modern slot machines have replaced mechanical reels with computer chips. A microchip known as a Random Number Generator (RNG) selects a number that corresponds to a specific reel position. The number of possible outcomes depends on the number of reels and symbols. For example, a slot machine with three reels and 25 symbols per reel has 15,625 possible combinations.

Casinos usually display the average rate of return on their machines. A machine that offers an 85 per cent return implies that, over time, a player might expect to regain £85 for every £100 staked. Key points about slot machines include:

• The RNG determines the outcome.
  
• A maximum coin wager is advised to achieve the highest jackpot.
  
• Changing machines frequently can help manage losses.
  

Video Poker

Video poker is similar to traditional poker, but the player competes against the machine. The game relies on the ranking of hands, with the royal flush at the top. Other hands include the straight flush, four of a kind, full house, flush, straight, three of a kind, two pair, and a pair of jacks or better.

Different machines offer varying payouts based on their pay tables. For instance, a machine that rewards a full house at 9–1 and a flush at 6–1 is more favourable than one with lower payouts. Some of the key aspects are:

• A clear ranking of poker hands is essential.
  
• The pay table determines the return and the house advantage.
  
• Higher payouts usually require the maximum coin wager.
  

Baccarat

Baccarat is a simple card game where the player chooses to bet on either the player's hand, the banker's hand or a tie. Two hands are dealt from an eight-deck shoe. The cards have a fixed point value: aces count as 1, tens and face cards count as 0, and the remaining cards hold their numerical value. When a hand totals more than nine, 10 is subtracted from the sum.

The rules for drawing an extra card are predetermined. A total of 8 or 9 on the first two cards is known as a natural, and no further cards are dealt. A table summarising the house advantage in baccarat is shown below:

Betting on the banker usually offers the best odds.

Texas Hold-Em Poker

Texas Hold-Em is a card game where players bet on the best hand formed from two private cards and five community cards. The game emerged in the 1960s and is now very popular in casinos and tournaments. The decisions made during the game rely on the probabilities of completing a strong hand.

Players calculate "pot odds" to decide if a call is worthwhile. The number of "outs" – cards that can improve a hand – also plays a key role. A short list of recommended starting hands includes:

• High pairs such as tens, jacks, queens, kings, and aces.
  
• An ace paired with a king, queen, or jack.
  
• King combined with queen or jack.
  
• Suited connectors like jack and ten.