Probability of Empty Intersection of Randomly Chosen Planes?

[Dragonfall]

Let $x \in \{-1, 1\}^n$ and let $p(x) = \{w \in \mathbb{R}^n : x \cdot w > 1\}$. What is the probability that $p(x_1) \cap \ldots \cap p(x_{n+1}) = \emptyset$ given that $x_i$ are chosen uniformly at random?

 

[voko]

$p(x)$ is not a plane. It is a half-space. If you insist on the symbolical formulation, then the question boils down to the probability of having at least two vectors in $\{ x_1, \ ... \ , x_{n + 1} \}$ that are anti-parallel.

 

[Dragonfall]

No, it's possible to have empty intersection without a pair of opposite vectors.