Probability that n hemispheres cover a sphere

Question

$n$ hemispheres of a sphere are chosen at random. What is the probability that they cover the sphere?

Details : The process of random selection of the hemispheres, although unspecified, might play a role in the final answer. Assume it is uniform with whatever sample space and random variable you choose.

My efforts so far : The equivalent problem in 2D with semicircles is equivalent to :

If a segment of length 1 is randomly divided into n intervals, with what probability are all intervals are less than 1/k?

With k =2 and the length of segment being $2\pi$.

Edit : The solution can be found here : https://www.cut-the-knot.org/m/Probability/HemisphereCoverage.shtml