Help with geometric probability (with regards to span). Suppose we have two non-parallel vectors in $\mathbb R^3$.
Now, if we were to randomly select another vector in $\mathbb R^3$, what is the probability that that new vector lies in the span of the first two vectors?
My intuition says that the probability is 0 (i.e. $\mathbb R^2$ is infinitely smaller than $\mathbb R^3$, ergo 0), but I'm not sure.

One way to model this, so everything becomes finite is to work on the unit sphere $S^2$. We may assume that the hyperplane is $H=\{z=0\}\subset \mathbb{R}^3$. The probability that a random vector of $S^2$ doesn't lie on the great circle $H\cap S^2$ would be exactly
$$
\frac{|S^2\setminus (H\cap S^2)|}{|S^2|}= 1. 
$$
The only problem here is that we exclude the zero vector, but I assume that this is not a big deal. So, I would say your intuition is correct.
