Find probability of remaining paint free - ball duel, Applied Statistics

Find probability of remaining paint free - ball duel, Applied Statistics

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In a three-cornered paint ball duel, A, B, and C successively take shots at each other until only one of them remains paint free. Once hit, a player is out of the game and gets no more shots. The three paint ballers have different probabilities of hitting their target. A hits the target 30% of the time, B hits the target 50% of the time and C (the brute) hits the target 100% of the time. A shoots first, followed by B, then C, then back to A if paint free, etc. If each player adopts the best strategy at each turn, including possibly an intentional miss, find the probability of remaining paint free for each of A, B, and C.

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