# Intersection of random planes

## Main Question or Discussion Point

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?