A CRASH COURSE IN GAMING MATH
Even though they might be unable to express it clearly in words, almost every gamer understands gaming math. For example, even though you might not know the exact possibility of getting a royal flush at a poker table, you certainly know that it’s really low.
This crash course in gaming math will introduce you to concepts that would turn the vague intuition you already have into practical knowledge backed by proof! You might already be aware of some of the ideas discussed here, but it’s never the wrong time for a quick refresher.
Before we dive into it, we should say that this article is more about games than math. Don’t worry, the information has been presented in a way so everyone can understand clearly, not just those who are interested in math. Go games! Go numbers! Go people!
GAMING AND MATH—WHY EVEN BOTHER?
Often we prefer not knowing the math behind the casino games we love. There are many reasons for this. Firstly, math can seem scary with its complex equations and numbers. Secondly, many people also think that learning the true mathematical nature of casino games takes away the magic.
When it comes to games like poker, where skill is a significant factor, the game’s math becomes more critical. Professional poker players dedicate insane hours to memorizing possible outcomes and card ranges to gain a competitive advantage.
Interestingly, the truth is that math is not scary at all and can be a faithful ally if only we dedicate some time to learning basic concepts that encapsulate the field of gaming math.
It can be vital to winning in games like poker, and it can help us make the best choices in games like roulette. Overall, having a rudimentary understanding of the most important mathematical concepts driving our favorite games is something that every gamer should aspire to.
EXPERIMENTS, EVENTS AND OUTCOMES
To understand the math behind casino games, we need first to look at them from the perspective of math. In gaming math, all casino games can be thought of as experiments that generate random events.
As a fundamental example, consider a roll of dice. When we roll a die, we conduct an “experiment” that will result in an outcome. The outcome would be random, but we are already aware of all the possible outcomes from the event.
The combined set of outcomes possible from a roll of dice is as follows: 1, 2, 3, 4, 5 or 6. This is called the “sample space” of the experiment. Similarly, in the case of draw poker, the experiment’s sample space includes every possible 5-hand combination that a player can expect from a deck of 52 cards.
Understanding the sample size in games like poker can be very helpful as they allow you to calculate the odds of particular events happening in the game.
For example, the total number of 5-card distinct poker hands is 2,598,960 (calculation explained in the next section), and the number of possible royal flushes in the entire sample space is 4 (one for each suite.) Therefore, the chances of getting a royal flush are 4 in 2,598,960 (or roughly 0.000154%). Yikes, that’s small.
By understanding the sample space of events in a game, we can apply probability theory to figure out the odds of being successful or unsuccessful in the said game. All events in a casino game can be classified into separate categories depending on their nature. For example, events can rely on each other or be independent of each other, exclusive or nonexclusive, elementary or complimentary.
Games of chance like roulette and dice are easiest to model, because:
- The sample space of all events is finite and known.
- A simple event generates a random outcome from the sample space.
When we start talking about games like poker, where there are other factors than just probability at play, gaming math can help you better understand the chances you take. Still, it is only a part of the overall playing strategy, including estimating other players’ hands, the likelihood of their bluffing, and other factors.
PERMUTATIONS AND COMBINATIONS
Two of the core gaming math concepts are permutations and combinations. Fortunately, it is also straightforward to understand them. Both of them relate to arranging a set of objects or numbers. With permutation, we’re not concerned with the order in which the objects or numbers are placed, but the order is relevant with combinations.
Let’s take a look at an example. In the previous section, we learned that the total number of possible hands in draw poker is 2,598,960. To arrive at that number, we use a simple formula for combinations.
FORMULA FOR COMBINATIONS
There are 52 cards in a standard deck, and we want to find out how many 5-hard combinations are possible out of all the individual cards in the deck. In mathematical notation, we can express the same problem as nCr = n!/[r! (n-r)!] where n is the total number of objects and r is the size of the combinations we want those objects to be in.
For this particular example, n would be 52 and r would be 5 since there are 52 cards in a deck and a poker hand consists of 5 cards in draw poker.
If you look at the equation again, you’ll notice an exclamation mark next to the letters. This represents a factorial which basically tells us to keep multiplying the number with descending integers. So, 3! = 3x2x1 and 5! = 5x4x3x2x1 and so on.
The factorial of a number also tells us the number of ways we can arrange those numbers without repetition. For example, suppose you have three cards, and you want to find the number of possible ways you can organize them.
3! (read as “three factorial”) is the answer (3x2x1=6). It is also the total number of permutations without repetition possible for the 3 cards (but more on that later.).
When we plug the numbers into the combination formula, we get the following:
52![5! X (52-5)!] (from nCr = n![r! (n-r)!])
When you solve the equation, the answer is 2,598,960, as noted above, representing the total number of combinations for 5-card poker hands from a deck of 52 cards.
UNDERSTANDING EXPECTED VALUE
Expected value is an integral part of gaming math. In a nutshell, it tells us the average gain or loss we would get if we made a bet repeatedly with the same odds. To calculate the expected value of a bet, we simply multiply the number of possible outcomes with the reward for each of them.
Here’s a quick example to understand the concept better. Suppose you are speculating on a coin toss repeatedly; every time it lands on heads, you get 1 cherry, and similarly, you lose 1 cherry every time it lands on tails.
Since the probability of the coin landing on heads and tails is ½, we can calculate the total average expected value of the bets as follows: 1*(½) + (-1)*(½)
If we solve that, we get 0. This is an example of neutral expected value games, where you will neither win nor lose if you keep playing for an indefinite amount of time. Games can also have positive or negative expected values, which correspond to winning or losing a certain amount over time.
Let’s consider an alternate game: you still bet on a coin toss and lose 1 cherry every time it lands on tails, but you gain 2 cherries every time it lands on heads. The expected value, in this case, would be 2*(½) + (-1)*(½). If we solve that, we get ½ or 0.5 which means that you can expect a positive expectation if you play the game for an indefinite amount of time.
Coming back to poker, it is possible to calculate the expected value of every decision you make if all the variables influencing the play are known, such as the player’s hole cards. While the game still retains an element of chance, it is possible to bet in situations where you have a positive expected value.
WHAT IS THE “HOUSE EDGE”?
You probably know that in the long run, the house wins. This is because casino games are designed to give the house an advantage over the players. While players can win large amounts in the short-term, the casino will still come out on top in the long-term.
Naturally, the house edge is a highly relevant concept to gaming math and strategy. The house edge exists due to the difference between the odds offered by the game and the “true” statistical odds of that event happening. This does not mean that casino games aren’t fair, as anything is possible in the short-term; the house-edge can be considered similar to the spread charged by bookmakers.
However, not all casino games share the same house edge, and understanding the house edge of a game can be extremely helpful for gaming strategy. One of the most famous examples used to demonstrate the differences in house edge is roulette.
Multiple variants of roulette are played worldwide, but the two major variants are American Roulette and French/European Roulette. Both games are almost identical, except they come with different house edges. Let’s take a look at why this is the case.
In American Roulette, the wheel has two zeros with 18 black and 18 red segments, but there is only one zero in French/European Roulette. Suppose the player bets on red in both games.
For American Roulette, the player’s expected value would be the number of favorable outcomes * reward + the number of unfavorable outcomes * reward.
American Roulette: (18/38)1 + (20/38)(-1) = -0.054 French/European Roulette: (18/37)1 + (19/37)(-1) = -0.027
So while a player can expect to lose 0.054 cherries per bet on an indefinite game of American Roulette, they would only lose 0.027 cherries per bet on an infinite game of European Roulette. The house edge in American Roulette is 5.4% ( 0.054 * 100) and in French/European Roulette is 2.7%(0.027 * 100).
Unfortunately, calculating the house edge gets more complex for games like Blackjack, but it is possible.
WHY UNDERSTANDING GAMING MATH ADDS UP
Even though most casino games offer recreational experiments with luck for most of us, understanding these games’ features can be helpful and interesting.
The benefits of looking through casino games from a statistical and mathematical perspective go beyond just winning more; it can also help us enjoy the games more and see games for what they are—the opportunity to take a chance and for a thrill.
Note this is a reference article to gaming math in general. It is for information and entertainment only (if you like math, lol). It is not related to, nor a reflection of, Chumba Casino, its views, its products, content, or its games.