The Probabilities of Gambling
By: Catalin Barboianu
Tags: Probability, Gambling Mathematics, Gambling, Odds And Betting, Games Of Chance

Games of chance are a popular entertainment option, offering glamour, thrills and the possibility of winning big prizes. The sparkling slots, the green baize of the tables, the spinning roulette wheel and the roll of the dice entice and seduce … but they also have another side, one that is often hidden to players and operators. We refer to the mathematics behind the games, the models and applications that underlie their processes, predictions, and strategies.

The field of gambling mathematics is a wide one, dealing with a range of applications, such as probability, statistics, topology, decision theory and numerical analysis. All games can be modelled mathematically and it is surprising to see how simple-rules games hide complex mathematical structures, generating a range of applications with practical results.

Of all the mathematical aspects of games of chance, that of odds/probabilities is that which arouses the most interest. The likelihood of winning or losing in a given situation or over the long run is vital information for gamers, and an essential element in their gaming decisions: which games to play and which not, how to act in a certain moment, and how to adjust strategy to the odds. This information is also crucial to the industry: Casinos and operators need to ensure their house edge; game developers must understand the probabilities in order to establish the viability of slots and their payouts.

In general terms, probability is the most rigorous measure of the possibility of an event occurring. Just as length measures distance and area measures surface, probability measures random events, providing the events fit into a certain mathematical structure. In other words, we can measure the possibility of the same three symbols occurring on a slot machine winning line – as we can quantify all the parameters of the machine in terms of probability, but we cannot calculate the “probability” of the sun not rising tomorrow, as this event doesn’t fit into such a structure.

All games of chance feature a range of events, each with its attached probability. The probability of winning a multiple roulette bet, the probability of going bust on the next blackjack card, the probability of getting a certain sum on the dice roll, or the probability of your opponents holding a higher card in the flop stage of a Hold’em game, are just isolated examples of events for which probabilities can be calculated. As it happens, possible gaming events are infinite in number: red to occur 3 times in a row once in 10 spins, red to occur 4 times in a row in 10 spins, and so on, n times in a row in m spins… in other words, infinity.

On the other hand, the possible gaming situations within one game are finite in number, although this number can be huge. Taking an example from poker, the number of all possible board configurations (your own pocket cards and the community cards) in the turn stage of a Hold’em game is 5,987,800, rising to 53,890,200 when all possible player numbers are taken into account. Each hand has hundreds of probabilities attached, related to events occurring in your own hand or in your opponents’ hands. To give another example, the total number of possible roulette placements for a multiple bet amounts to 2^154, a 47-digit number!

What winning probabilities will we encounter in games of chance? The range is very wide, with odds of close to zero up to odds of 90%. Take for instance the following multiple bet at American roulette: a $16 bet on red along with 15 straight bets of $1 on 15 black numbers. The probability of winning (any bet) is about 87%. Pretty high, right? If the player wins the colour bet, he makes a $1 profit. If he wins a straight bet, the profit is $5. The profit rate is a small one, but the winning odds are very high. This strategy would appear to be suitable for a player who wants to accumulate small “sure” profits over the long run. If so, one may ask, where is the house edge? Are a few players running bets like this one going to ruin the house? Well, the answer is no, because there is another parameter that we haven’t taken into account — the possible loss. In the event the player doesn’t win any bet (neither the colour bet nor the straight bet), he will lose $31, which could easily ruin his previous gain. In mathematical terms, we may say that the expected value of this bet is negative.

The expected value (EV) is the predicted future gain or loss and is the sum of the probability of each possible outcome multiplied by its payoff. Accordingly, it represents the average amount one would expect to win when identical bets are repeated many times. In our example, the expected value is -$1.62. Accordingly, a player running this bet could expect to lose on average $1.62 for each $20 played. Thus, even though the winning odds are 87%, the EV is negative, ensuring the house isn’t ruined. Of all the games of chance, roulette can offer the highest winning probabilities, the above example being relevant in this regard. In roulette, the winning probabilities of some multiple bets (assuming they are non-contradictory, i.e., that the possible profit cannot be negative or zero) can reach 92%.

At the opposite end of the scale, lottery offers the lowest winning probabilities. For instance, in a 6/49 lotto, hitting all 6 winning numbers with a one-line ticket offers odds of 0.000007151%; hitting 5 numbers offers odds of 0.00184%, and the odds of hitting 4 numbers are 0.09686% — pretty close to zero. Still, even players who are aware of the figures continue to play, and lotto players are among the most perseverant gamblers: this has to do with the entertainment feature of the game.

In other games, we encounter probabilities spanning the entire range. In blackjack, for instance, the odds of winning range from bad to relatively good. To take a few examples: the odds of initially being dealt a hand of 18, 19, 20, or 21 points is 27.6% in the case of a 1-deck game and 27.7% in case of a 2-deck game. In the middle of the game, assuming we are playing with one deck and you are the only player at the table, suppose you hold Q, 2, 4, A (totalling 17 points) and the dealer’s face-up card is a 4. The probability of hitting 21 points with the next card is 4.25%, the probability of hitting 20 points is 8.51%, and the probability of hitting 19 points is 6.38%. Thus, the overall probability of hitting 19, 20, or 21 points is 19.14%.

At slots, a game with no opponents, the odds depend on the structure of each machine, i.e. the number of reels, the number of symbols on each reel and the winning combinations. For instance, in a 3-reel machine with 15 symbols, the probability of getting a specific double (a certain symbol twice) is 1.24% and that of getting any double is 18.66%. Slots offer a range of combinations, all measurable in probability.

Probabilities are particularly useful when making strategic decisions — not only in gambling, but also in other fields and in daily life. However, probability calculation is hard enough for a nonmaths person and available software is not of much help (most odds calculators are based on partial simulations rather than explicit probability formulas, and deliver statistical results rather than odds/probabilities. Many people — maths inclined or not — often rely on intuitions or “hunches” when estimating probabilities. However, hunches are often just illusory reactions and are not analytically grounded.

Probability is one of the domains where hunches can play bad tricks. Therefore, offering probability results remains the job of the applied mathematician. Of course, the results are not to be taken as an absolute degree of belief in the occurrence of gaming events, because mathematical models are ideal and abstract. However, probability is the only scientific measure we have for uncertainty, and must be used without distortion.

Dr. Catalin Barboianu is gaming mathematician, author of several books on gambling mathematics.