Expected value of area

Question

Let points $A_1, A_2, \ldots A_n$ be independent and have uniform distribution in a unit circle $B$. Let random set $Q$ consists of all points of circle $B$ such that: each point of $Q$ is closer to the center of circle than to its boundary and to each of $A_i$ points. I need to find expected value of area of set $Q$.

Answer 1

The approach for solving this has already been provided in the comments.

For a point $P$ at distance $r$ from the centre, the probability of a given one of the $A_i$ to be more than $r$ away from $P$ is $1-\frac{r^2}{R^2}$, where $R$ is the radius of the circle. Thus the probability that $P$ belongs to $Q$ is $\left(1-\frac{r^2}{R^2}\right)^n$. Integrating this over the circle yields the expected area of $Q$:

$$ 2\pi\int_0^Rr\mathrm dr\left(1-\frac{r^2}{R^2}\right)^n=\mathrm\pi R^2\int_0^1\mathrm d\frac{r^2}{R^2}\left(1-\frac{r^2}{R^2}\right)^n=\frac{\pi R^2}{n+1}\;. $$