A short course in mathematical expectation. This scenario underscores the importance of understanding mathematical expectation and the house edge, concepts that are crucial for anyone looking to navigate the complexities of casino games.

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There is a game of chance – roulette. The rules are as follows:

- There is a circle divided into 38 parts.
- Of these, 18 are black.
- 18 are red.
- One or two zeros are green.
- A player bets money on something from this circle.
- The croupier pops the balloon. The ball spins round the roulette wheel and falls on one of the fields.
- If the player guesses where the ball will fall, he takes his bet and some money on top.
- If not, his bet goes to the casino.

There are a lot of betting combinations, so we will consider the most popular bet on red or black. All other types of bets and their results are counted according to the same scheme.

It seems that with such payouts you can always be in the plus: after all, it is enough to guess the colour, and it falls out in almost half of cases. But it’s the other way round: people lose more often than they win. Let’s find out why this happens.

## Probability of events

Let’s assume that we throw an ordinary dice with numbers from 1 to 6. The probability of falling out one is β , because all sides of the die are the same and fall out randomly.

This can be represented as simple maths:

If we have several equally possible and identical events, then the probability of any one of them occurring is 1/n, where n is the number of such events.

## Mathematical expectation

If we take a strict definition and write it in simple words, it turns out like this:

The mathematical expectation is when we add up the products of the probabilities of each event by their outcomes.

This means that the mathematical expectation is the average result we will get each time we try to play the game. The more attempts we make, the closer our result is to the mathematical expectation.

Let’s explain it with the example of a dice.

We know that the probability of each edge falling out is β , and the numbers on the die are from 1 to 6. We rolled the first time: we got 6. The second time, 1. Then 4. Then 2. Then 5. And so on. Can we predict what the average result will be after a hundred or two games?

As it turns out, you can. Knowing only the probability and the number of points on each side of the die, we can say in advance what will be the average result of throwing this die if we throw it long enough. This is calculated using the formula:

( β Γ 1) + ( β Γ 2) + ( β Γ 3) + ( β Γ 4) + ( β Γ 5) + ( β Γ 6) = 21/6 = 3,5

The more times we roll the die, the closer our average will be to this number.

## Rolling the die for money

Knowing the mathematical expectation can help us make the right decision in all sorts of gambling, betting and financial matters.

Imagine a game like this: you are offered to roll a dice and get as many dollars as the number of dollars rolled on the die. The price of one roll is three dollars. Is it worth playing such a game or not?

From the point of view of expectation – yes, it is worth it, and here is why:

- We know that the expectation at each roll of the die is 3.5. In our game, this means that the average win per roll after a notional 1000 rolls will be $3.5.
- Since the expectation is higher than the cost of one throw, we should not just agree to such a game, but play this game as long as possible to eventually reach the average profit of $0.5 per throw.

It is possible to throw the die 10 times in a row so that only 1, 2 or 3 falls on it – and then it is as if we are at a disadvantage. But if we play this game long enough, we’ll be at a gain.

The main thing to remember is that the mathematical expectation does not guarantee that we will get this exact result on the first try. We might not get it on the tenth. But if we keep trying long enough, then we will definitely get closer to the desired result.

## Expectation and roulette

Now that we know the theory, we can calculate the expectation for playing and betting on roulette.

Many people think that if you bet on red or black, you have a 50/50 chance of winning or losing. But it’s not. It’s all about the zero – it creates a negative mathematical expectation for the player, and it is easier to lose than to win. Now, watch the numbers carefully.

If we have one zero, we have 18 black + 18 red + zero = 37 cells in roulette. Suppose we bet on red, then our chance of winning is 18/37 = 48.6%. It turns out that we have a higher chance of losing – 51.4%. The difference is 2.8%.

If we bet a thousand dollars on each bet, this difference in chances gives us a negative mathematical expectation of 28 dollars not in our favour. This means that on average with each bet we lose $28 from every thousand dollars.

The interesting thing is that we do not lose them every time: there is no such thing that after every bet someone went into our pocket and took out 28 dollars. But if some mass of people make some mass of bets, then in the end, according to the amount of money they have left on their hands, they will see a loss of 2.8%. But since they will not count each other’s money, they will not know it.

Casinos make money precisely because of the zero, which creates a negative expectation for the player. If there are two zero sectors in roulette, it doubles the expectation in favour of the casino.

To make money, casinos don’t need to cheat, load roulette, make magnetised balls, etc. Just having one stray sector that creates a negative expectation is enough.

## Betting on a number

To be clear, let’s look at another betting option where a player bets on a specific number or even a zero and when he wins, he gets 35 times what he bet.

Most likely, you can already see what the catch is: the bet is 35 times more, and the sectors are at least 37. This means that if you bet a thousand dollars, the expectation will be 35/37 Γ 1000 dollars = 945 dollars. It turns out that in this case the player loses even more than just choosing a colour – $55 vs $28.

If you play roulette for a long time, because of the negative expectation, the player will gradually lose money until they run out. The longer he plays, the more he will lose.

## But someone sometimes wins, right?

There are casino wins, but they are random events that cannot be guaranteed to happen again. For example, you can bet a large sum on a number and accidentally win 35 times that amount and walk away. It is in the hope of such an eventuality that people go to casinos.

But with smaller wins, people want to increase profits, and in pursuit of the next luck lose all the money on bets, including the money won. And all because the more bets a player makes, the stronger the expectation works in favour of the casino and the faster he will lose everything he came with.

But the casino always earns as long as there are a lot of people in its hall. Lots of people, lots of bets. With negative expectation the casino earns predictably and always, while its customers earn randomly.

## And one more thing: expectation and budgets

Mato-expectancy is a pretty picture, but there’s a catch. Counting the results of hundreds or thousands of games makes sense if you have an unlimited budget. If you have a limited budget, you may not have enough to use expectation.

Let’s say you come to the only casino in the world that gives you a positive expectation. But there is a restriction: the minimum bet is $10,000. What next:

- If you have $100,000 in your pocket, you will run out of money after 10 losing bets in a row. You will not even have time to use the expectation.
- If you have a million in your pocket, you will have enough for 100 lost bets. 100 is on the edge. Maybe you will start to feel the effects of expectation, maybe not.
- To be sure, you need to come to the casino with 10 million in your pocket. Even so, with a +2% expectation in your favour, you will leave the casino with a profit of 200,000. If you put the same 10 million on deposit, you would at current rates would get a profit of 300 thousand.
- And this – in a non-existent casino with a positive expectation for you. There are no such casinos in nature.

In short: keep your money in savings banks. The casino always makes money. The expectation is a heartless bastard.

## So, can you beat the casino?

It is possible to win at a casino. The probabilities of winning here are higher than in the lottery, but still they are calculated in units and fractions of per cent. The main rule – go to the casino with the money that is not sorry to lose, and if you win – stop. But you can not beat the casino.