Single-sample distribution in Simon's algorithm

If the hidden string is zero, one oracle-Hadamard run samples every $y\in {\mathbb{F}}_2^n$ with probability $2^{-n}$. If the hidden string is nonzero, it samples every $y$ satisfying $y\cdot p=0$ with probability $2^{-(n-1)}$ and every other string with probability zero. Pairing $x$ with $x\oplus p$ makes the forbidden amplitudes cancel.