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Take a stick of unit length and break it into three pieces, choosing the break point at random. (The break points are assumed to be chosen simultaneously). What is the probability that the three pieces can be used to form a triangle?

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Answer:

We can form a triangle if none of the resulting pieces is at least 1/2, i.e. if no ... It is not hard to show that if you choose a point randomly in this triangle, the distances to the three ... of lengths that you obtain by breaking a stick at two random points. ..... we get line segments joining the midpoints of the edges of the triangle.

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