Example — Working out the odds to set

Let us reverse the problem: given the probability and the desired margin, how much should the bet pay out? It is enough to impose the wanted expected value and solve for the net winnings.

Example — Working out the odds to set

Suppose the probability of the event “Zverev wins and Sinner wins” is $0,2$. A betting house wants to secure a margin of $10\%$ on the $100$ EUR bet (that is, $\bar V = -10$). What odds should it set for the event?

One sets up the equation with unknown net winnings $x$: $0,2\cdot x + 0,8\cdot(-100) = -10 \quad\Longrightarrow\quad x = \frac{70}{0,2} = 350 \text{ EUR}.$ The net winnings must be $350$ EUR; returning also the $100$ EUR staked, the house pays $450$ EUR for every $100$ EUR wagered: odds of $4,5:1$.

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Topics: Probability
Concepts: Fair game · Expected value
Methods: Fair game · Expected value
Skills: Probability calculation · Solving equations