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I mean that [itex]x+V[/itex] intersects the axis [itex]\langle e_n\rangle[/itex] iff there exists [itex]\alpha\in\mathbb{R}[/itex] such that [itex]\alpha e_n\in x + V[/itex] and [itex]\alpha \neq 0[/itex].

By "likely" I mean that for example if [itex]x[/itex] is a sample point of some random vector, and if [itex]V[/itex] is spanned by some [itex]M[/itex] sample vectors, and if the random vectors in question can be described by non-zero probability densities, then what is the probability for intersections of [itex]x+V[/itex] and [itex]\langle e_n\rangle[/itex] to exist?

Example M=1, N=2. If we draw a random line on the plane, chances are that the line will intersect both axes. The outcome that line intersects only one axis, is a special case, which can occur with zero probability.

Example M=2, N=3. If we draw a random plane into three dimensional space, chances are that the plane will intersect all three axes. The plane can also intersect only two or one axes, but these are special cases, which can occur with zero probability.

Example M=1, N=3. If we draw a random line into three dimensional space, changes are that the line will miss all three axes. The line can intersect one or two axes, but these outcomes occur with zero probability. The outcome that the line would intersect all three axes is impossible.

Looks complicated! I don't see a pattern here. What happens when [itex]N>3[/itex]?