Q: Peter throws two darts at a dartboard, aiming for the center. The second dart lands farther from the center than the first. If Peter now throws another dart at the board, aiming for the center, what is the probability that this third throw is also worse (i.e., farther from the center) than his first? Assume Peter's skillfulness is constant.

S: Since the three darts are thrown independently, they each have a 1/3 chance of being the best throw. As long as the third dart is not the best throw, it will be worse than the first dart. Therefore the answer is 2/3.

Ranking the three darts' results from A (best) to C (worst), there are, a priori, six equiprobable outcomes.

possibility # 1 2 3 4 5 6 1st throw A A B B C C 2nd throw B C A C A B 3rd throw C B C A B A

The information from the first two throws shows us that the first throw will not be the worst, nor the second throw the best. Thus possibilities 3, 5 and 6 are eliminated, leaving three equiprobable cases, 1, 2 and 4. Of these, 1 and 2 have the third throw worse than the first; 4 does not. Again the answer is 2/3.

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