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 $x_1,x_2,\ldots,x_n$ and probabilities $P(X=x_i)=P_i$ the mathematical expectation is:

$$E(X)=\sum_{i=1}^n x_i\cdot P(X=x_i)=x_1\cdot P(X=x_1)+x_2\cdot P(X=x_2)+$$ $$+ \ldots +x_n\cdot P(X=x_n)$$

Four people are betting $1€$ 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 $1€$ is $\dfrac{5}{6}$, since we will lose if the selected number is not the result.

On the other hand, the probability of gaining $3$ $€$ is $\displaystyle \frac{1}{6}$.

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

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.