The Probabilistic Survivor

Discrete Mathematics Level 5

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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