Suppose we are betting. We can either win or lose a certain amount of money at each bet. We can play two strategies, strategy $A$ with a winning ratio of $a$ and expected value $\alpha$, and strategy $B$ with winning ratio $b$ and expected value $\beta$, where:
\begin{equation} a = \frac{\text{bets won, while playing with strategy $a$}}{\text{total number of bets played}} \end{equation}
and
\begin{equation} \alpha = a\cdot(\text{gain from winning bet})+(1-a)\cdot(\text{loss from losing bet}) \end{equation}
(losses are negative values). $b$ and $\beta$ are defined in the same way. My question is: knowing the values of $a,b,\alpha$ and $\beta$, and knowing that
\begin{equation} a>b\qquad\qquad\text{and}\qquad\qquad\alpha<\beta \end{equation}
which strategy should we play (in order to maximize gains) if we can only play once?
My gut feeling says strategy $a$ even if the expected value is greater for the other strategy. But I don't know how to mathematically prove it (or disprove it).