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PMO National Stage Oral Phase

Geometry Difficulty 4.7 AIME Prove it Philippines

Problem:
A school program will randomly start between $8{:}30$ AM and $9{:}30$ AM and will randomly end between $7{:}00$ PM and $9{:}00$ PM. What is the probability that the program lasts for at least $11$ hours and starts before $9{:}00$ AM?

Solution

Solution:
Consider a rectangle $R$ with diagonal having endpoints $(8.5, 19)$ and $(9.5, 21)$ . Let $S$ be the region inside $R$ that is to the left of the line $x=9$ and above the line $y=x+11$ . The desired probability is given by
$\frac{\text{ area of } S}{\text{ area of } R} = \frac{\frac{5}{8}}{2} = \frac{5}{16}$

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.