A unit length, thin rod breaks in two places, each break independently uniformly distributed on [0, 1). Let a be a small number. Show that the probability that one of the three pieces into which the rod breaks has length less than a is approximately ka as a → 0 where k is a constant you should determine.

A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
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Chapter1: Combinatorial Analysis
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A unit length, thin rod breaks in two places, each break independently uniformly distributed on
[0, 1). Let a be a small number. Show that the probability that one of the three pieces into which
the rod breaks has length less than a is approximately ka as a → 0 where k is a constant you
should determine.
Solution. The relevant regions of the unit square are the four strips of width a along each edge
and a strip of width a/V2 around the X1 = X2 diagonal (which is of length v2). Ignoring
overlaps (since as a → 0 they will contribute quantities depending on a²) they have a total area
of 6a.
Transcribed Image Text:A unit length, thin rod breaks in two places, each break independently uniformly distributed on [0, 1). Let a be a small number. Show that the probability that one of the three pieces into which the rod breaks has length less than a is approximately ka as a → 0 where k is a constant you should determine. Solution. The relevant regions of the unit square are the four strips of width a along each edge and a strip of width a/V2 around the X1 = X2 diagonal (which is of length v2). Ignoring overlaps (since as a → 0 they will contribute quantities depending on a²) they have a total area of 6a.
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