The Probabilistic Survivor

Three mortal enemies $$A$$, $$B$$ and $$C$$ decide to settle their private feuds through a triangular 'duel'. They stand at equal distances from each other and shoot to kill. They all agree to continue shooting in a certain order $$A C B$$ until only one survivor is left. Each of them knows that $$A$$ kills his target with probability $$\frac{1}{4}$$, $$B$$ kills his target with probability $$\frac{3}{4}$$ and $$C$$ kills his target with probability $$\frac{4}{4}$$.  However, $$A$$ decides to change the order of shooting so as to maximize his survival probability. His proposed order is unwittingly accepted by $$B$$ and $$C$$ and they then agree to shoot in this order.  The difference between the survival probability of $$A$$ in the order finally followed and in the initial order $$(A C B)$$, can be expressed as $$\frac{a}{b}$$. What is the value $$a + b$$?  $$\textbf{Details and Assumptions:}$$

• $$A, B$$ and $$C$$ always adopt optimal strategies so as to maximize their individual survival probability.

• They are allowed to shoot once per turn and can shoot anywhere they want.

• Nobody dies due to a shot not intended for him.

• Upon being hit by a bullet, the shot person dies immediately

This is part of Ordered Disorder.

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