Mathematical expectation

The mathematical expectation $E (X)$ is the sum of the probability of every possible event multiplied by the frequency of the mentioned event, that is to say if we have a discrete quantitative variable $X$ with $n$ possible events $x_{1}, x_{2}, \dots, x_{n}$ and probabilities $P (X = x_{i}) = P_{i}$ the mathematical expectation is:

$E (X) = \sum_{i = 1}^{n} x_{i} \cdot P (X = x_{i}) = x_{1} \cdot P (X = x_{1}) + x_{2} \cdot P (X = x_{2}) +$ $+ \dots + x_{n} \cdot P (X = x_{n})$

Example

Four people are betting $Math input error$ on a number of a dice, each chooses a different number. Then for each euro that they have bet, if winning, they get $3$ euros. Is it worth betting in this game?

The probability of losing $Math input error$ is $\frac{5}{6}$ , since we will lose if the selected number is not the result.

On the other hand, the probability of gaining $3$ $Math input error$ is $\frac{1}{6}$ .

This way the expectation is: $E (X) = (- 1 \cdot \frac{5}{6}) + (3 \cdot \frac{1}{6}) = \frac{- 5}{6} + \frac{3}{6} = - \frac{2}{6} = \frac{- 1}{3} ≃ - 0.33$

Therefore, for every euro bet we can lose $0.33$ cents. It is said that this is a game of negative expectation.

We say that a game is equitable when the benefit expectation is $0$ .

If we have a game expectation of positive benefit, it is said that it is a game of positive expectation.