Alice and Bob play chess against each other. Alice wins in probability $\frac 34$ if Charlie bets on her and in probability $\frac{1}{3}$ if Charlie bets on Bob. Charlie gets a 1$ if he's right and loses one if he's wrong. Charlie would like to determine a range of bets guaranteeing him a revenue. Determine if such range exists and if it does determine the optimal bet.
So far, denoting $A=\{\text{Alice wins}\},G_A=\{\text{Charlie bets on Alice}\}$ I found that since $$ P(A)=P(A\mid G_A)P(G_A)+P(A|G_A^c)P(G_A^c)=\frac 13+\frac{5}{12}P(G_A). $$ Now I take great pains finding the expected value of the revenue. Am I even on the right track?