Do planes in multi-dimensional spaces rarely intersect?

Question

Am I correct in saying two planes in 5-dimensional space will typically intersect in a point? Or will they not even intersect not at all?

Answer 1

The plane $\{(x,y,1,0,0)\}$ and the plane $\{(0,0,0,z,w)\}$ have empty intersection. This is typical, in the sense that in any reasonable parametrization ("moduli space") for such pairs of planes, the set of pairs with empty intersection is open and dense.

Answer 2

You could think of it this way: in five dimensions, to specify a plane via equations of the form $ax_1 + bx_2 + cx_3 + dx_4 + ex_5 = f$, you require three equations.

To specify the second plane is another three equations.

You now have six equations in five unknowns. What is the chance they have a solution? That is how likely you are to find a point of intersection of the two planes.