Crash probability in half an hour from the hourly rate

Question

At a certain junction, the chance of one or more crashes during a given hour is $\tfrac89$ . Crashes happen independently at a steady rate over time. What is the chance of one or more crashes during a $30$ -minute window?

Short answer

$2/3$ . An hour with no crash means two independent crash-free half hours, so $(1-p)^2=\tfrac19$ and $p=\tfrac23$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

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

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