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Chapter 13: Problem 11

Three contestants, \(A, B,\) and \(C,\) each has a balloon and a pistol. From fixed positions, they fire at each other's balloons. When a balloon is hit, its owner is out. When only one balloon remains, its owner gets a \(\$ 1000\) prize. At the outset, the players decide by lot the order in which they will fire, and each player can choose any remaining balloon as his target. Everyone knows that \(A\) is the best shot and always hits the target, that \(B\) hits the target with probability \(.9,\) and that \(C\) hits the target with probability \(.8 .\) Which contestant has the highest probability of winning the \(\$ 1000 ?\) Explain why.

Short Answer

Expert verified
Overall, A has the highest probability of winning the prize, with a chance of approximately 0.167.

Step by step solution

01

Determine the shooting order

As the shooting order is determined by lot, every player has 1/3 chance to shoot first, second, or third.
02

Analyze possible scenarios if Player A shoots first

Player A always hits the target. This means that if A shoots first, he will definitely hit someone's balloon. The logical choice for A would be to shoot B, as he is the second best shooter, leaving the weakest shooter, C, in the game. If A shoots first and hits B, we are left with a duel between A and C, where A has a 100% chance to hit the balloon and C only 80%. In this duel, A will win the game if he gets to shoot first, which happens in 50% of the cases. Therefore, the chance of A winning if he shoots first is 1/3 (chance to shoot first) * 1/2 (chance to win the duel) = 1/6.
03

Analyze possible scenarios if Player B shoots first

If B shoots first, he will choose to eliminate A from the game because A is the best shooter. If B manages to hit A's balloon (which happens with probability 0.9), it's then a duel between B and C. In this duel, B will win if he shoots first, which happens 50% of the time. Therefore, the chance of B winning if he shoots first is 1/3 (chance to shoot first) * 0.9 (chance to hit A's balloon) * 1/2 (chance to win the duel) = 0.15.
04

Analyze possible scenarios if Player C shoots first

If C shoots first, he will also aim at A because A is the best shooter. If C manages to hit A's balloon (which happens with probability 0.8), then there's a duel between B and C. In this duel, C will only win if B misses, which happens with a probability of 0.1. Therefore, the chance of C winning if he shoots first is 1/3 (chance to shoot first) * 0.8 (chance to hit A's balloon) * 0.1 (chance B misses) = 0.027.
05

Sum up the probabilities of winning for each player

Adding up the probabilities from each of the scenarios, we find that A has a probability of 1/6, or approximately 0.167, to win the game; B has a probability of 0.15 to win; and C has a probability of 0.027 to win.

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