#spurious-features

Inputs are vectors $x \in \{-1, 1\}^d$ with uniformly random coordinates, and labels $y \in \{-1, 1\}$. For each coordinate $k$ there is a model that predicts either $x_k$ or $-x_k$, whichever fits the training set better. On $n = 10$ training examples, what is the probability that the model for a coordinate that is irrelevant to the label (independent of $y$) has zero training error?