A short course in mathematical expectation.

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

- There is a circle divided into 38 parts.
- 18 of them are black.
- 18 – red.
- One or two green zeros.
- The player bets money on something from this circle.
- The croupier launches the ball. The ball spins on a roulette wheel and falls on one of the fields.
- If the player guessed where the ball will fall, he takes his bet and some money from above.
- If he guessed wrong, his bet goes to the casino.

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

If the player bets on a color – red or black – he gets doubled back. If he bets on a specific number, he gets 35 times more than he bet.

It seems that with such payments you can constantly be in the black: after all, it is enough to guess the color, and it falls out almost half of the time. But the opposite is true: people lose more often than they win. Let’s see why this is happening.

**This has already happened**

We have already talked about the expectation that solved the problem about the foot-po-li-hundred . In short:

- We are considering some possible future events.
- The probability of these events is described by a number. For example, 1 – the probability is 100%, the event will definitely happen. 0.5 – the event occurs on average in half of the cases.
- If the event is associated with some kind of win or loss, we use simple mathematics to assess the profitability of a particular game.
- This number that describes profitability is called the mathematical expectation.

Now let’s look a little deeper.

**Probability of events**

Let’s say we roll a regular dice with numbers from 1 to 6. The probability of getting one is ⅙, because all sides of the dice are the same and fall out randomly.

This can be thought of as simple math:

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

**Expected value**

If you take a strict definition and write it in simple words, it will look like this:

Expectation is when we add the products of the probabilities of each event to their outcome.

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

Let us explain with the example of a dice.

We know that the probability of each face being rolled out is ⅙, and the numbers on the die go from 1 to 6. We rolled the first time: it fell 6. The second time – 1. Then 4. Then 2. Then 5. And so on. Is it possible to predict What will be the average result after a hundred or two games?

It turns out you can. By knowing only the probability and the number of points on each side of a die, we can tell in advance what the average roll of that die will be if we roll it long enough. This is calculated by the formula:

(⅙ × 1) + (⅙ × 2) + (⅙ × 3) + (⅙ × 4) + (⅙ × 5) + (⅙ × 6) = 21/6 = **3.5**

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

It turns out that the expectation shows what result we get, on average, **if we play the game fairly USD th** .

**Roll the dice for money**

Knowing the expected value can help us make the right decision in all kinds of gambling, disputes, and financial affairs.

Imagine a game like this: you are offered to throw a dice and get as many rubles as the dice. The price of one throw is three rubles. Is it worth playing such a game or not?

From an expectation point of view – yes, it is, and here’s why:

- We know that the expectation for each die roll is 3.5. In our game, this means that the average winnings for a throw after a conditional 1000 throws will be 3.5 rubles.
- Since the expectation is higher than the cost of one throw, we need not just agree to such a game, but play this game as long as possible in order to eventually reach the average profit of 0.5 rubles per throw.

You can roll the dice 10 times in a row so that only 1, 2 or 3 will fall on it – and then we seem to be in the red. But if we play this game long enough, we will win.

The main thing to remember is that expectation does not guarantee that we will get exactly this result on the first try. Maybe we won’t get it with the tenth. But if we continue these attempts long enough, then we will definitely get closer to the desired result.