What is the probability that the point lies on the circumference of a (centered) circle of radius r/2 inside the circle with radius r?

Question

A point is chosen at random inside a circle of radius $r$. Find the probability that the point lies on the circumference of a (centered) circle of radius $r/2$ inside the circle with radius $r$ ? What I tried was that I drew another circle of radius $r/2$ inside the circle of radius $r$. Dividing circumference of the circle (radius $r/2$) by the area of the circle (radius $r$) you get probability $1/r$. Is it correct?

Answer 1

Short answer: The probability of hitting a 1-dimensional curve in a 2-dimensional space will (usually) be zero.

Assuming a regular distribution (in particular no deltas) on the circle of radius $r$, the probability of the random point lying exactly on the circumference of a circle of radius $r/2$ is zero. This is because the circumference has measure zero (being a smooth one-dimensional curve of finite length in $\mathbb R^2$).

Further reading: See What is the Lebesgue measure of a circle in $\mathbb{R}^2$ and Is the Lebesgue measure of the unit circle $0$? for explanations why the boundary of a circle has zero Lebesgue measure.