Accident in half an hour given the hourly chance

Question

Watching traffic from a bus stop, you know that the probability of seeing at least one accident during a one-hour window is $\tfrac34$ . Assuming accidents are equally likely at any time and the two halves of the hour behave independently and identically, what is the probability of seeing at least one accident in a $30$ -minute window?

Short answer

$\tfrac12$ . If $q$ is the chance of no accident in half an hour, then two independent halves give $q^2=1-\tfrac34=\tfrac14$ , so $q=\tfrac12$ and the answer is $1-q=\tfrac12$ .

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
Continuous Random Variables and Probability Density Functions
Cumulative Distribution Functions

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