TITLE: Probability that random subspaces intersect
QUESTION [5 upvotes]: Given an ambient space $\mathbb{R}^d$ and two randomly oriented subspaces $A,B$ with dimensions $a,b$ respectively, how can I express the probability that $A$ and $B$ intersect non-trivially?

REPLY [3 votes]: Assuming a uniform independent distribution (say picking orthogonal vectors to span each space one at a time by uniform distributions on spheres) the probability is zero for $a+b\le d$ and one for $a+b>d$.
To see why, take $A$ to be fixed by rotation to some standard space. Then consider picking $b$ orthogonal vectors to span $B$. Note that a nontrivial intersection is equivalent to linear dependence amongst these two bases. But the probability the first introduces it is zero unless $A=\mathbb R^d$ because the angle a vector makes with a hyperplane is 0 with probability 0. Clearly the answer is one in the $a=d$ case.
But then instead of immediately picking the second vector orthogonal to the first, simply project onto the orthogonal complement. $A$ retains the same dimension, but now $b\to b-1,d\to d-1$. Thus we have the result above by induction.