Odds of 2,5 to 1 and the margin

Text

A bookmaker offers odds of $2,5:1$ on an event $E$ with $P(E)=0,3$. (a) What is the average winnings when betting $50$ EUR? (b) What are the fair odds? (c) What percentage margin does the bookmaker keep for itself?

Solution

(a) With odds of $2,5:1$ a stake of $50$ EUR, if the event occurs, yields a net win of $1,5\cdot 50 = 75$ EUR (you collect $2,5$ times the stake, of which $1$ is the returned stake). The expected value is $\bar V = 0,3\cdot 75 + 0,7\cdot(-50) = 22,5 - 35 = \boxed{-12,5\ \text{EUR}}$ The bet is therefore unfavourable to the player.

(b) The fair odds give $\bar V = 0$. Solving for the net win $x$: $0,3\cdot x + 0,7\cdot(-50) = 0 \;\Rightarrow\; x = \frac{35}{0,3} = \frac{350}{3} \approx 116,67\ \text{EUR}.$ The corresponding odds are $(50 + 116,67)/50 \approx \boxed{3,33:1}$.

(c) The margin is the expected loss relative to the stake: $\frac{12,5}{50} = \boxed{25\%}$

Links

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