This article is from the Puzzles FAQ, by Chris Cole firstname.lastname@example.org and Matthew Daly email@example.com with numerous contributions by others.
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 skilfulness is constant.
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