I'd like to find a closed form (if possible) expression of the probability of interesection of two geometrical figures $F_1$ and $F_2$ of area $A_1$ and $A_2$, respectively, that are have a random position and orientation in a bounded 2-dimensional space of area $A_{tot}$.
Obviously, this probability depends on the exact geometry of $F_1$, $F_2$, and the space in which they live. However, is there a closed form expression of this probability for some classes of geometries or shall I go for Monte-Carlo methods?
Thank you very much!
Greg