Target surviving a randomly placed blast

Question

A target sits at the centre of a circle of radius $1$ mile. A bomb lands at a uniformly random point in this circle and destroys everything within $\tfrac12$ mile of where it lands. What is the probability that the target survives?

Short answer

$\tfrac34$ . The target is hit only if the bomb lands within $\tfrac12$ mile of the centre, a disc with $\left(\tfrac12\right)^2=\tfrac14$ of the area. So it survives with probability $\tfrac34$ .

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

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Lessons behind this question

Continuous Random Variables and Probability Density Functions
Cumulative Distribution Functions
Sample Spaces, Events, and the Probability Axioms

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