Shortest of three random stick pieces

Question

A one-metre rod is cut at two points chosen independently and uniformly along its length, producing three pieces. Find the probability that the shortest piece is no longer than 0.2 m.

Short answer

$0.84$ . The complement is that all three pieces exceed 0.2 m. Removing 0.2 from each piece leaves a uniformly random split of length $0.4$ , so this has probability $(1-3\cdot0.2)^2=0.16$ , and the answer is $1-0.16$ .

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Lessons behind this question

Sample Spaces, Events, and the Probability Axioms
Conditional Probability, Total Probability, and Bayes' Rule

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