Casino – the product of mathematicians, and brilliant. Almost all the mathematical basis for the functioning of gambling institutions appeared a long time ago and is relevant to this day. Everyone knows that the institution always remains in the “plus”. But why? It seems that yesterday you won and withdrew money.

In fact, everything has long been calculated and based on conventional probability theory – as, however, and the work of your favorite games: whether it’s slots, roulette or poker. The theory of probability studies the regularities of random phenomena. That’s what it says on the great and mighty Wikipedia.

How can a casino make a profit?

But what does it mean in the field of gambling?

The outcomes of all gambling, including online slots, are random. I mean, they can’t be predicted in advance. We’re gonna need the mathematical laws of probability theory, which are actually quite simple. First we need to understand that any spin in the slot machine is subject to certain laws.

Yes, we do not know what will be the result of a separate rotation, but we can fairly accurately predict the profit, for example, after 100,000 spins. The theory of probability in online casinos consists of many theories, laws and other established knowledge. The key for us will be the following concepts:

In principle, we do not need anything else to describe the work of the entire casino machine. And this, for a moment, a complex and huge body! In general, let’s take it in order.

## Mathematical expectation

In gambling it is important to approximate your potential profit. Mathematical expectation is exactly what helps to do this. It shows the average of a random value.

What is a random value in the same slot machines? This is the win of the individual back! Thus, mathematical expectation shows the average of our winnings.

And if you have any knowledge of modern slots, you know very well about the existence of such an indicator as the percentage of payouts (Return to Player). It, in fact, will be our mathematical expectation. The payout percentage shows how much the slot will give back for a long period of time.

This means that the machine is programmed to give at a distance (ie, after a huge number of spins) 96% of the stake invested in it. Not in every spin, but after a large number of them. These same 96% can be called the expectation of a particular game. Only the expectation is not expressed in percentages, it can be in the range of -1 to 1. So our 96% transform into -0.04. And therefore the mathematical expectation is negative. On average, on a distance, it would mean 4% loss.

For example, it’s 96%.

#### How do you get the positive mathematical expectation in the casino?

Mathematical expectation can be positive, just in the case of the casino it happens very rarely.

it can happen in blackjack (when counting cards), but in the Internet with this is much more difficult (because the deck is infinite). Or even when playing with a profitable bonus – say, 200% with an x25 Vader.

For example,

Negative mathematical expectation, ie, the potential profit of the player – this is absolutely standard for almost all casino games.

In fact, the player can not make a profit on the distance (but can win a lot on the short run), so do not try to make your hobby the only source of income. It is rather an entertainment for which you have to pay.

## Random magnitude dispersion.

Dispersion is a measure of deviation of a random value from the mathematical expectation. Roughly speaking, it is a measure of deviation from the average value.

take the Space Wars slot. It is considered highly dispersive, which means a large deviation of random variables from the mathematical expectation. In the player’s language, it sounds like, “Eat like a vacuum cleaner!” “Grizzly” slot because it can make big winnings. And how else can large payouts be balanced, if not by long periods of plums?

For example,

In Space Wars (conditional) player can get a maximum of x5,000 per spin. But the law works in the opposite direction, so the pool of maximum losses (ie, a series of empty spins) is also lengthened. In another slot machine, the maximum win can be x1,000 for spins. Consequently, it is much less dispersive. There is one important point. The higher the deviation from the mathematical expectation, the more “non-standard” outcomes will be. For example:

x100 win,**result 1:**10 empty spins,*result 2:*x10 win**result 3:**win x2,*result 4:*20 empty spins*result 5:*

and so on. Even after hundreds of spins we may not approach the average of a random value – the mathematical expectation. The same Space Wars gives 96.8% at a distance. If you want, you can roughly estimate whether your results match this value. Although you don’t have to worry about it – most likely it is not. You may be in a big “plus” or a big “minus”, but it is unlikely that you played zero even after a large number of spins.

Why not? Because the dispersion is high.

## Normal distribution.

Normal distribution – distribution of probabilities that directly depends on the values of mathematical expectation and dispersion. It is most likely that you have encountered it more than once before. This is the same bell-shaped graph, the highest point of which (peak) corresponds to the value of the mathematical expectation (average value).

In other words, the normal distribution describes the spectrum of scatter of random outcomes. In our case, it will be roulette rotations, spins in slots or rolls of dice in the craps. The higher the dispersion, the lower will be the maximum peak of the chart. Conversely: the lower the variance, the higher the maximum value.

In case of Space Wars automatic machine, the chart of its normal distribution is similar to a small hill. This can be explained by the large dispersion of all outcomes. For example:

x50 win (we put a dot on the chart to the right),*result 1:*10 empty rotations (draw points to the left),*result 2:*win x200 (point still right),*result 3:*win x2 (points almost in the middle of the chart),*result 4:*win x2,000 (point almost on the right edge).*result 5:*

and so on. Of all the outcomes (points) and formed a chart. Most of the spins give insignificant winnings and losses and therefore are concentrated in the center. But all the big winnings and numerous periods of constant plums ** “flatten”** the graph, as if to stretch it to the sides. All random variables are subject to normal distribution, if they form a large array of weakly dependent data. And here, by the way, we can mention one more important mathematical law.

#### Central limit theorem

The central limit theorems generally state the following: if we have a large sample of almost unrelated random variables and none of them is dominant, then their distribution will tend to normal.

There are several central theorems, but they all talk about the same thing. In the casino, all of this also applies and works perfectly.

## Large numbers

The law of large numbers is, in fact, synonymous with limit theorems. It states: if there are a large enough number of random, equally distributed values, their average value tends to their mathematical expectation. It’s the same, only in a little different words.

As you can see, the key concepts in casino probability theory are the concepts of mathematical expectation and variance, and all axioms, theorems and laws are formed around them.

The law of large numbers is universal and works in many areas, not just in the casino. In the philistine sense, it demonstrates the possibility of a positive outcome on an extremely large array of random data. Often given as an example of the probability of life on Earth. If the probability of its appearance of 1 of a billion planets, and in the conditional galaxy of 30 billion planets, then another 30 of them have life. But this is a very controversial application of this law.

## Mathematical expectation of casino bonusesâ€¦

It is quite interesting and useful to calculate the expectation of specific bonus offers. It is quite simple, for this we will need the following values:

*a vager,**casino advantage in a bonus wagering game,**the bonus amount.*

So, ** first we take a Vager (let’s say it equals x50) and multiply it by the value of the bonus (for example $ 100) to find out the total sum of bets. We got $ 5,000 (50 * 100 = 5,000). I.e.** we need to place a wager of $5,000 to win back the 100% first deposit bonus, which is quite standard. Next, we multiply $ 5,000 by the advantage of the institution in a particular game. Most likely it will be some kind of slot, in machines NetEnt return is most often about 96%, so the advantage of the casino 4% : 5 000 * 0.04 = $ 200. It is we got the theoretical value of loss after putting $ 5,000 in bets. This is very approximate, most often the practical value is very different.

*Further from the bonus we take away the theoretical loss and get a mathematical expectation: 100 – 200 = -$ 100. It turns out that it is not profitable to take the bonus.*

In the same way you can calculate the “profitability” of any bonus offer. But note that this is all theory, in most cases you will have completely different results in practice.