This problem is inspired by multiple battle royale games. In the games, there is often a boundary which closes in around players, forcing them towards a specific point so that the last players remaining must fight.
The question I have is: There is a unit circle centered at the origin of the Cartesian Plane (representing the outer boundary of the game map). Another circle of radius $a$ is centered such that it is completely inside the unit circle. It is placed randomly and fairly. Given a point $(x,y)$, what is the probability that is is within the smaller circle? How does the probability change if more circles are nested inside each other with radii of $b,c,d...$? What happens if different shapes are used?
Finally, how could I solve similar problems in the future?
The area of a circle is proportional to the radius squared (in fact the constant of proportionality is $\pi$, formula $A=\pi r^2$). So the ratio of areas is:
$$\frac{a^2}{1^2}=a^2$$
which is also the probability.
The answer would be the same for other similar figures, for example a square of side length $1$ containing a smaller square of side length $a$.
With multiple figures, it is just:
$$\frac{a_i^2}{a_j^2}=\left(\frac{a_i}{a_j}\right)^2$$
where $a_i$ and $a_j$ are the radii (or whatever characteristic length is used) of the two figures you consider.