Choosing a radius in a laser-pointer bet

Question

A circular target has radius $10$ , and Mordecai shines a laser at a point chosen uniformly over its area. Before he does, you pick a number $r$ with $0\le r\le 10$ . If the dot lands within distance $r$ of the centre, you pay him $10-r$ dollars; if it lands farther out, he pays you $r$ dollars. Which $r$ gives you the largest expected profit?

Short answer

$r=5$ . You lose $10-r$ with probability $p=r^2/100$ and win $r$ otherwise, so the expected profit is $r-p\cdot 10 = r-\tfrac{r^2}{10}$ . This downward parabola peaks at $r=5$ .

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

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