Let N points be uniformly distributed on the surface of a unit sphere $S^2$. What is the probability that every spherical cap of area A contains at least one point?
The area $A$ depending on the angle $\Phi$ between the tip of the spherical cap and the boundary of the cap.
The uniformly distributed points are generated by generating Gaussian random vectors in $R^3$ with i.i.d. entries and normalizing them.
EDIT. The same problem in 2D case. Consider N points being distributed uniformly over the circumference of a unit circle (N can be large enough). Let any arc subtend an angle $\theta$. For a given arc, the probability that a point falls inside the arc is $p = \frac{\theta}{2 \pi}$. The probability that the point falls outside the arc is $q = 1 - p$. Probability that all points fall outside the arc is $q^N$ which is same as the probability that no point falls inside the arc. Now the probability that at least one point falls inside the arc is $1 - q^N$. So far so good. This result is for one particular arc.
My question is what is the probability that every arc of angle $\theta$ contains at least one point? I am unable to generalize from one arc to every arc. Similar treatment in 3D case from one particular spherical cap to every spherical cap.