How close to the sprinkler?

Question

A sprinkler at the center of a circular lawn of radius $9$ m wets the lawn evenly, so a wet point is uniformly random over the disk.

What is the probability that a given wet point is within $3$ m of the center?

Accepted Answer

Probability inside a disk scales as the square of the radius: P(	ext{within 3 m}) = \left(\frac{3}{9}ight)^2 = \left(\frac13ight)^2 = \frac19 . Takeaway. A radius that is one third of the way out captures only \left(	frac13ight)^2 = 	frac19 of the area. Area near the center is deceptively small.

How close to the sprinkler?

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