Q11E Three contestants, A, B, and C, ... [FREE SOLUTION]

Chapter 13: Q11E (page 501)

Three contestants, A, B, and C, each has a balloon and a pistol. From fixed positions, they fire at each other鈥檚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 inwhich they will fire, and each player can choose anyremaining balloon as his target. Everyone knows that A is the best shot and always hits the target, that B hitsthe 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

Contestant C has the highest probability of winning the $1000 because none of the other players will try to eliminate it first due to its comparatively lower probability of hitting the target.

Step by step solution

Finding the contestant with the highest probability of winning the $1000

In this game, each contestant will try to eliminate their strongest opponent to increase their chances of winning.

A has the highest probability of hitting the target, followed by B and C. Contestant A will decide to shoot B first to improve the chances of winning because C is more likely to miss shooting A than B if B is eliminated first.

B will try to eliminate A first because its chances of winning against C are higher.

Similarly, C will try to eliminate A first because its chances of winning against B are comparatively higher. Both B and C will benefit from shooting A first; A has the least chance of winning.

Thus, if A goes first, it will shoot B鈥檚 balloon, but both B and C will shoot A鈥檚 balloon if either goes first. Neither A nor B will shoot C鈥檚 balloon first, thereby increasing its chances of winning. Therefore, C has the highest chance of winning the prize money worth $1000.