You break a stick at two points chosen at random, and thus obtain three pieces. What is the probability that these three pieces can form a triangle?
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The broken stick
Classic geometric probability: two random slices, a central region, and an answer that seems too small until you see it.
Hints
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- It is not necessary to compare each piece with the sum of the other two.
- Just look at the longest piece.
- The three pieces form a triangle if and only if the longest is less than half the length of the stick.
Solution
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The condition for three segments to form a triangle is that the longest one is less than the sum of the other two. Since the sum total of the three pieces is the length of the stick, that is equivalent to saying that the longest piece must measure less than half the length of the stick.
Now we represent the two breaking points within a square of possibilities. The favorable region is the central area in which no piece exceeds half of the stick.
Drawing that region yields a central triangle whose area is exactly one-fourth of the total area of possibilities. Therefore, the probability sought is 1/4.
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