Will Archer A Survive?
Three archers, A, B, and C, are standing equidistant from each other, forming an equilateral triangle. Archer A, B, and C has , , and 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.
The answer is 0.397.
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1 solution
The optimum strategy for weak player A is to deliberately miss his first shot. The optimum strategy for players B and C is to take a shot at each other on their turns. Lets prove how.
Lets take player A. He deliberately misses his shot. Now B takes his shot at C.
1) Scenario 1 - B misses the shot with 1/3 chance.
C on his turn shoots B dead. A shoots at C can now survive with a 1/3 chance. Obviously if he misses he is dead. Total chance of survival = 1/3 * 1/3 = 1/9
2) Scenario 2 -- B shoots down C with 2/3 chance.
Now its between A and B. A shoots B dead with 1/3 chance on first round.
But if A misses, he still has a chance if B misses as well. So A misses with 2/3 chance, B misses with 1/3 chance, then A shoots B dead with 1/3 chance in the next turn. This can continue infinitely setting up a geometric series in the bracket
1/3 { 1 + 2/9 + (2/9)^2 + (2/9)^3......} which sums up to 1/3 *9/7 = 3/7.
This is multiplied by 2/3 so its 2/7.
Total Probability of both scenarios = 1/9 + 2/7 = 25/63 = 0.397...
Now to prove that A missing first shot deliberately is best strategy.
1) A takes the first shot at B.
a) In this case, if A hits B he is a goner since C will hit A with certainty.
b)Therefore, the only way A can win is if he misses but in this case, the situation is just as it was before, when A shot to miss. Since White has a 2/3 chance of missing, and from above we know that A has a 25/63 chance of winning if he misses, this shows that the probability of A winning is 2/3 * 25/63 = 50/189 in this case, or roughly 26.5%
2) A takes first shot at C.
a) If he misses the shot, The scenario is the exact same as above and he wins with a 50/189 chance.
b) If he shoots C dead (1/3), A can only win if B misses his shot (1/3), and then A wins the face-off against B (3/7). So its 1/3 * 1/3 * 3/7 = 1/21.
Therefore the total probability that A will win if he shoots at C is 50/189 + 1/21 = 59/189, which is roughly 31.2%...