Expected profit from true and payoff odds

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.

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