Closer to the center than to the rim

Question

A point is chosen uniformly at random inside a disk of radius R. What is the probability that the point is closer to the center than to the boundary circle?

Accepted Answer

Let r be the point's distance from the center. Its distance to the boundary circle is R - r. "Closer to the center" means r < R - r \quad\Longrightarrow\quad r < \frac{R}{2} . So the point must lie in the inner disk of radius R/2, which has probability \left(\frac{R/2}{R}\right)^2 = \left(\frac12\right)^2 = \frac14 . Takeaway. Being nearer the center than the rim is not a coin flip: it needs r < R/2, and that inner disk is only a quarter of the area.

Closer to the center than to the rim

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