To know how exactly we should split the scene, we need some assumption:

rays are randomly distributed

rays are not occluded

Theorem: for convex object A inside convex object B, the probability that a random ray that hits B also hits A is given by the ratio of the surface area S_A and S_B of these objects.

Pr\{\text{hit } A | \text{hit } B\} = \frac{S_A}{S_B}

Therefore, given a split, we can compute the cost of using this split: