I had posted this as a problem but I think my approach to the problem wasn't right.
Three archers A , B and C are standing equidistant from each other, forming an equilateral triangle. Archer A , B and C has 1/3 , 2/3 and 1 probability of hitting the target they aimed for , respectively.
The three archers will play a survival game. The objective of the game for all players is to kill the other two archers and be the only survivor. Deliberate missing is not allowed.
The person who is to shoot a particular shot is to be chosen AT RANDOM among the people alive right before the shot ( which means ABAABBBABAA is possible as is CAAAAA ) .
Assuming that all archers will die if he is hit by an arrow aimed at him, and that all archers will make the best possible optimal moves to maximize their chances of winning (surviving) the probability that Archer A wins can be expressed as a/b where a and b are co-prime positive integers.
All comments are welcome...