Working in the set of Real Numbers.
Say that you're given a set of basis vectors $\{\mathbf{b}_i\}, \, i \in [n]$. Now, sample a vector v uniformly at random from the entire space. What is the probability that v can be written as a linear combination of any $n-1$ basis vectors? More specifically, is it finite or negligible?
Any references on how to go about this would be great too.
EDIT: The vector v has unit magnitude.
The first issue here is that sampling a vector uniformly at random from $\mathbb R^n$ isn't well-defined; you can't have a uniform distribution when your space is unbounded like this.
However, assuming any reasonably distribution, the probability is zero. The probability of your chosen vector being $0$ in the first coordinate is zero, as is the probability of being $0$ in the second coordinate, and so on. The probability of at least one of these happening is at most the sum of the probabilities of any of them happening, which is simply zero.