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.