Example — Odds and the house margin

The odds of a bet never reflect exactly the probability of the event: the difference is the house’s guaranteed profit. This example makes it quantitative.

Example — Odds 1,5:1 and the house margin

A betting house offers odds of $1,5:1$ on the event $B$ (“Zverev beats Medvedev”): betting $1$ EUR and winning, you receive $1,5$ (of which $0,5$ EUR is net winnings and $1$ EUR is the return of the stake). Suppose $P(B)=0,6$.

I bet $100$ EUR. If I win I pocket $50$ EUR net, if I lose I lose $100$ EUR. $\bar V = 0,6\cdot 50 + 0,4\cdot(-100) = 30 - 40 = -10 \text{ EUR}.$ The expected value is negative: on average I lose $10$ EUR for every $100$ EUR bet, that is, the house keeps a margin of $10\%$.

Fair odds. For the game to be fair we would need $\bar V=0$, that is $0,6\cdot x + 0,4\cdot(-100)=0$, from which $x = 100\cdot 0,4/0,6 \approx 66,\overline{6}$ EUR. The fair pricing would therefore be about $1,66:1$ (and not $1,5:1$).

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