= Coverage of a sphere by random open hemispheres
{title2=$1-\frac{N(N-1)+2}{2^N}$}
The probability that $N\geq1$ <independent> uniformly oriented open <hemispheres> cover a <sphere> is the displayed quantity. Coverage fails exactly when their normals lie in a closed hemisphere. With probability one, no three normals are coplanar and no pair is antipodal; a closed hemisphere containing the normals can then be perturbed into an open one. Condition on the unoriented axes of the normals. Their $N$ orientation signs remain independent fair bits, and the <regions of a general-position great-circle arrangement> realize exactly $N(N-1)+2$ of the $2^N$ possible sign vectors. Each such vector corresponds to normals that fit into one open hemisphere, proving the complementary probability. The argument counts equiprobable orientation choices, not equal-area regions.
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