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Sept. 24, 2003
Truel in the Sun
Agent Math, while journeying through the far reaches of Abstraction, meets her two nemeses: Robin Hood's deranged twin brother and an evil centaur. In order to settle once and for all which one of them is the greatest, they agree to stand in a triangle and fire, in turn, at one of the others until only one remains standing.
Robin Hood's brother, unfortunately, is an excellent shot, and he hits his target every time. The centaur is pretty good and hits his target two-thirds of the time. Agent Math hits her target only one-third of the time.
Because Agent Math is the worst shot, the truelists agree that she should shoot first, followed by the centaur, then Robin Hood's twin.
At whom should Agent Math fire?
—Emma Pierson, Grade 7
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Each truelist has the option of firing into the air instead of at an opponent.
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Agent Math should fire into the air.
She certainly doesn't want to shoot at the centaur. If she's unlucky enough to hit him, Robin Hood's brother will get her on the next shot.
If she fires at and hits Robin Hood's brother, then the centaur will have the first shot against her. His overall probability of winning the duel will be 6/7 and hers only 1/7. Not very good odds.
If Agent Math misses on purpose, she will have the first shot against either Robin Hood's brother or the centaur in the next round. With probability 2/3, the centaur will hit Robin Hood's brother, and she will have an overall winning probability of 3/7. With 1/3 probability, the centaur will miss Robin Hood's brother, in which case Robin Hood's brother will get rid of his stronger opponent, the centaur, and your overall chance against Robin Hood's brother will be 1/3.
By shooting in the air, Agent Math's probability of winning the truel is 25/63 (about 40 percent), the centaur's probability is 8/21 (about 38 percent), and Robin Hood's brother's is only 2/9 (about 22 percent).
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