Die with winnings and losses

Text

I roll a die. If it shows $6$ I win $30$ EUR, if it shows $5$ I win $6$ EUR, if it shows $1,2,3$ or $4$ I pay $5$ EUR. Is the game fair? If not, what is the average win/loss per roll?

Solution

Each face has probability $1/6$. The expected value is the average of the payoffs weighted by the probabilities: $\bar V = \frac{1}{6}\cdot 30 + \frac{1}{6}\cdot 6 + \frac{4}{6}\cdot(-5) = 5 + 1 - 3,33 = \boxed{2,67\ \text{EUR}}$ The game is not fair: the average win is about $2,67$ EUR per roll, so it is favourable to the player (a fair game would have $\bar V = 0$).

Links

Topics: Probabilita
Concepts: Gioco equo · Valore atteso
Methods: Valore atteso
Skills: Calcolo probabilita · Modellizzare
Exercise type: Problema modellizzazione