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

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