If one randomly cuts the Plane ($\mathbb{R}^2$) into pieces along three (uniformly distributed) straight lines, then with probability 1 this leads to 6 pieces of infinite size and one triangle piece (excluding the case of parallel cuts, as they have a probability of 0). In the same way, cutting along four straight lines leads to one quadrilateral piece.
But for five (and more) cuts, this is not guaranteed to happen. 5 cut example with no pentagon piece
So what is the probability of 5 random cuts leading to a pentagon. And is this even a well defined problem?
Because a constant probability density over $\mathbb{R}^k$ would have to be 0 everywhere to not diverge, but then this would also lead to a total probability of 0 instead of 1. And if I try to start with a bounded shape in which I randomly place the cuts, so that i can define the probability distribution for the cut placement. Than taking the limit of larger areas to get the plane case would probably lead to different values depending on the starting shape like here.