# 400 probability/darts.p

## Description

This article is from the Puzzles FAQ,
by Chris Cole chris@questrel.questrel.com and Matthew Daly
mwdaly@pobox.com with numerous contributions by others.

# 400 probability/darts.p

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.

probability/darts.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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