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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.

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)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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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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