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.