y suggestion was to perform a RAM deletion* on the machine, and then to study at what point the divergence in the counters appeared and why. To my surprise, however, my friend seemed upset by my proposal , considering nothing short of heresy the notion of erasing the memory of a machine for the sole purpose of studying how its counters behaved . My friend argued that performing a deletion of this kind would interrupt the machine cycle , thus altering its behavior and generating greater uncertainty on future results . It is worth mentioning that this friend of mine is a connoisseur of casino games and slot machines in particular , has over twenty years’ industry experience , and has successfully occupied almost every conceivable position within a casino right from paying assistant to director of operations.
In recounting this particular anecdote, I wanted to illustrate a certain bias that has long existed, at a subconscious level, among many of us in the industry – I call it the “casinero syndrome” -, namely a tendency to put to one side all our technical knowledge and instead put our faith in mystical beliefs or theories, in which many of our players also blindly believe. While my friend was no doubt aware that his theory concerning altering the machine cycle contradicted all that we had learnt together for years, I'm sure to this very day he has not had a change of heart and considered performing a RAM deletion.
As such, for the sake of my good friend and for those of you who, like us, have not received formal training in statistics and probability, , I want to take a look at the basics regarding how casino games in general and slot machines in particular generate profit for a casino . I also want to address some myths and misconceptions around the topic. In this first article , we will explore the concepts of ‘mathematical advantage’ and ‘expected value’, which in subsequent articles will be followed by a look at notions of probability and calculating game parameters .
* Deletion of RAM (Random Access Memory) is a procedure performed to remove data residing in the physical memory of a machine, which are stored, among other placess, in the machine’s counters.
ADVANTAGE MATH (HOUSE EDGE)
While gambling has been practiced from time immemorial – there being evidence of this in various ancient cultures , such as the Babylonians , Sumerians , Assyrians, Chinese, and even in Greek mythology – it was not until the mid-seventeenth century that the concept of a ‘banked game’, as we know it today, emerged.
Previously, including in the Middle Ages, gambling had been considered " fair and just ", with no participant enjoying any advantage over the rest . Punters were generally subject to the same set of rules and the same randomness as to the final outcome of a game; therefore , there was a relationship of "fair balance" between gamblers wagering their possessions on the occurrence or nonoccurrence of a randomly determined outcome of a game .
This changed in the seventeenth and eighteenth centuries, when the concept emerged in Europe of House Banking (Bank or House), where a dealer or an establishment offers the possibility of multiple gamblers, all gambling against the Bank, thereby foregoing the " fair balance " that existed previously. These new games have the feature of ensuring a certain mathematical advantage for the Bank, called House Edge, whereby it is more likely for a portion of the money wagered to end up in the coffers of the Bank than with the players. This is the concept that gives rise to modern day casino games.
To achieve this mathematical advantage, modern games use one of the following mechanisms:
• Special rules that can only be used by the Bank and not by punters .
• Game options for possible random outcomes that are reserved only for the Bank.
• Rules that ensure that the Bank is more likely to win than the bettors .
• A disproportionate ratio between the total bet size and the prize money on every game.
• A relationship of disproportion between the options for game results that can be covered by bettors and the prize money awarded.
the mathematical advantage for the Bank ( House Edge) is ensued through special rules governing the game, and not by manipulating the behaviour of the instrument that is used to define the random results of the game.
As we saw in the previous paragraph, the specific rules built into games affords a casino a long-term advantage called ‘House Edge’, determined by the nature of the rules and with a large number of players being offered the chance to win a grand prize in the short term.
The expected value ( Expected Value or EV for short ) is the magnitude that allows us to determine the percentage of income the long term in relation to the total amount of money wagered by players on every play . The expected value, then, is the estimated winnings of the Bank for a certain game , provided the results of that game are in line with the statistical predictions.
Strictly speaking , the expected value can be defined both from the perspective of the casino and the player. That is to say, a positive expected value for the casino will be negative value for the player, and vice versa; if any game situation was beneficial for the player then for this player the expected value would be positive, and negative for the casino. The greater the magnitude of the expected value , the greater the expected gain or loss for the casino in the long run for that game .
Without delving into the more complex math, we can define the expected value of a certain position or situation of a game as the sum of the product of the probabilities of the different possible alternatives for their respective benefits or costs . In other words , the expected value is the average gain or loss resulting from a game situation considering all possible outcomes and their probabilities . We can express this in a practical way using the following equation :
EV = (Money that can be won X Probability of winning) + ( Money that can be lost X Probability of losing)
If EV > 0 the expected outcome of the game is a gain .
If EV < 0 the expected outcome of the game is a loss.
If EV = 0, the expected outcome of the game is a draw (neither a win nor a loss) .
In our next article on this we will look at some examples of calculating the expected value of casino games, which will help the reader get a clearer idea of the concept. Very briefly we can say that for slot machines the expected value is what we call “the Hold” (or retention rate ) . For those interested in learning more about these themes, I would recommend the website ‘Eye in the Sky’, which features an extensive set of examples of these concepts applied to different gambling machines.