Probability in Gambling

Probability is the likelihood that something will happen. It is measured as a number between zero and 1, where zero denotes impossibility and 1 denotes certainty. In other words, the higher the probability, the more certain the event will happen: a nice thing to know when gambling.

The probability that event X will occur is expressed as P(X). The probability that event X will not occur is expressed as P(not X) = 1 - P(X), and is known as the complement or opposite of event X. If two events X and Y occur together, it is referred to as intersection or joint probability of X and Y, indicated as P(X and Y). Throughout this article, we'll incorporate casino-related examples to illustrate how probability influences gambling.

Casino Applications

Probability is the essence of gambling. Casinos contain fixed conditions with random outcomes, so your chance of winning is easily calculable. The catch is that payout rates tend to correlate with your odds of winning; the better your odds, the lower the payout.

In any event, casinos tend to pay out slightly lower than the inverse probability. For example, if you have 1-in-5 chance of winning your bet (probability = 0.2), the casino may only payout 4:1 (instead of 5:1). The difference is referred to as the house edge, and it is how casinos and gambling sites make their money.

This article focuses on probability, but you should always factor in the payout and house edge when deciding whether or not to place a bet. A decent shot at winning does not translate to a worthwhile wager. For a list of common casino games and their respective rules, head over to our games page.


Common Games: Craps

Probability is usually easy to calculate in dice games. Each die can yield six outcomes (1-2-3-4-5-6). The probability of any single number landing on a six-sided die is 1/6, but as the number of dice increases the probabilities are no longer equal. For the purpose of this article, we'll look at Craps which uses two dice.

When two dice are used, there are 36 possible combinations that can result in any of 11 numbers between two and twelve. However, not every number has a probability of 1/11. For instance, there are six combinations that will produce the number seven (1-6, 2-5, 3-4, 4-3, 5-2, 6-1), making a 7 the most likely number to be rolled (probability = 1/6). Nevertheless, betting on the number seven in Craps carries a substantial house edge and the bet is best avoided.

If we look at the most popular bet, the Pass Line Bet, the probability is two-fold. The first roll, called the come-out roll, establishes the point. The exception is if a 2, 3, 7, 11 or 12 are rolled. A 7 or 11 will result in an automatic win. Since these numbers represent mutually exclusive events, it is called the union of these events and the probability is expressed as P(7 or 11) = P(7 U 11) = P(7) + P(11). We established that the probability of rolling a seven is 6/36. There are two combinations (5-6, 6-5) that will produce an eleven, which gives a probability of 2/36. So the probability of winning your wager on the come-out roll is P(7) + P(11) = 6/36 + 2/36 = 8/36.

If a 2, 3 or 12 are rolled, you automatically lose your bet. The probability of losing is thus P(2) + P(3) + P(12) = 1/36 + 2/36 + 1/36 = 4/36. Any other number establishes the point, and the game continues. This is the most likely result (probability = 36/36 - 8/36 - 4/36 = 24/36, or 2/3). Once the point is established, subsequent rolls take place until the result is either the point (win your bet) or a seven (lose your bet). At this stage, no matter what the point may be you are more likely to lose your bet than win.


Common Games: Poker, Blackjack, Baccarat

Card games such as Blackjack and Baccarat are fairly straightforward, since only the value of the card matters, and not the suit. Games such as Poker are much more challenging when it comes to calculating probability. To simply things, we will look strictly at the probability of the hand you are dealt.

When a single card is dealt, the value and suit are not mutually exclusive. The probability is expressed as P(V or S) = P(V) + P(S) - P(V and S). For example, when you are dealt a single card from a normal deck of cards, the odds of that card being a face card (J-Q-K), a spade, or both is 12/52 + 13/52 - 3/52 = 11/26, since the 52-card deck has twelve face cards, thirteen spades, and three that are both. The three that are both are included in each the twelve face cards and the thirteen spades, but you should only count them once.

If you were looking specifically at a single card being dealt, say the Ace of hearts, you would apply joint probability: P(A and H) = P(A)P(H). Thus, your odds of your first card being the Ace of hearts are 4/52 x 13/52 = 1/52. Once a card is dealt, it is no longer available as an outcome and must be subtracted from the total remaining cards. For instance, if you were dealt four cards that contained one Ace and four hearts, the chance of being dealt another Ace OR heart would be 3/48 + 9/48 = 1/4. In this case, we do not have to account for the Ace of hearts since it has already been dealt.

Numbers and Symbols

Common Games: Roulette, Slots, Keno, Bingo

The most basic gambling probabilities come in the form of numbers and symbols. You simply divide the number of desired events by the number of total events. For example, the chance of a European Roulette wheel landing on black would be 18/37. If you want calculate the chance of hitting the jackpot on a slot machine, you would apply joint probability. A three-reel slot machine with ten symbols per reel would give you 1/10 x 1/10 x 1/10 = 1/1,000 chance of hitting the jackpot. The same basic principles can be applied throughout the casino.

In conclusion, probability is a key component of both live casinos and online gambling sites. If you would like to learn more, check out another article or head to our homepage to try your luck at the top-rated worldwide gambling websites. Always remember to please gamble responsibly, and never risk more money than you can afford to lose.