# Will Archer A Survive?

**Discrete Mathematics**Level 4

Three archers, A, B, and C, are standing equidistant from each other, forming an equilateral triangle. Archer A, B, and C has \(\frac { 1 }{ 3 } \), \(\frac { 2 }{ 3 } \), and \(\frac { 3 }{ 3 } \) probability of hitting the target they aimed, 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. The order of shooting will be in alphabetical order (A, B, then C). Assuming that all archers will die if he is hit by an arrow aimed at him, and that all archers will make the best moves possible to maximize their chances of winning (surviving), what is the probability that archer A will survive and win?

Round your answer to the nearest thousandth.

**Details and Assumptions**:

- The archers are allowed to skip their turn if they want to. If so, there is a 0 probability chance of hitting any target.