Circular distribution of circles

Question

Suppose you have $n$ objects , distributed randomly, in a circular manner of radius $r$. Each objects is of area $A$. So my question is if you draw line everywhere from the center to the surface of the circle, what fraction of all the 360 degrees lines will reach the surface untouched by the objects?

Answer 1

Either the objects can't overlap in this case it's just like putting them side by side. The blue circles have an area $A=\pi a^2 $, $a$ is its radius. The angle $\theta$ verifies $sin(\frac{\theta}{2}) = \frac{a}{r}$ since the red triangle is isosceles. The probability of having a radius of the green circle not touching any blue circles is : $$p=\frac{360-n\theta}{360}$$

Otherwise the objects can overlap, in this case each object represents a new trial, each trial being independent. The probability of having a radius of the green circle not touching any blue circles is : $$p=\left(\frac{360-\theta}{360}\right)^n$$