The true odds against event $E$ are $r$ to $1$ and the payoff odds are $t$ to $1$. You bet an amount $B$:

  • You win $tB$ if the event occurs.

  • You lose $B$ if the event does not occur.

How do you calculate the net winnings?

Calculating the probabilities of winning and losing, I got $\frac{1}{r+1}$ and $\frac{r}{r+1}$, respectively. Hence,

$$\text{Net Winnings} = \frac{tB}{r+1} - \frac{r B}{r+1} = \left(\frac{t-r}{r+1}\right) B$$

Does this seem correct? Any help would be appreciated. I'm not sure where to start.

  • $\begingroup$ @RodrigodeAzevedo, is the probability of losing r/(r+1)? $\endgroup$ – H Bn Jun 17 '18 at 19:33
  • $\begingroup$ Note that the expected profit is zero when $r = t$. $\endgroup$ – Rodrigo de Azevedo Jun 17 '18 at 20:30
  • $\begingroup$ @RodrigodeAzevedo thank you! $\endgroup$ – H Bn Jun 17 '18 at 22:31
  • $\begingroup$ Related: favorite-longshot bias. $\endgroup$ – Rodrigo de Azevedo Jun 20 '18 at 8:17

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