orientations $\bf a$, $\bf b$ of the two analyzers on Figure 1 may be different. If $\theta$ is the angle $\angle(\bf a,\bf b)$ between vectors $\bf a$ and $\bf b$ then the two outcomes $(+1,+1)$ and $(-1,-1)$ in which the two measurements give us the same result, have probability $\frac12\cos^2\theta$ each, and the other two outcomes $(+1,-1)$ and $(-1,+1)$ have probability $\frac12\sin^2\theta$ each [@Peres §6-2]. In particular, if $\bf a = \bf b$ then the probability of getting the same result is 1.

To illustrate how Bell’s Theorem leads to negative probabilities, David Schneider played with three orientations $\vec A, \vec B, \vec C$ in his blog post [@Schneider]. The angle $\angle(\vec A, \vec C) = 3\pi/8$, and $\vec B$ is in between so that the angle $\angle(\vec A, \vec B) = \pi/4$ and $\angle(\vec B, \vec C) = \pi/8$. (We swapped Schneider’s $\vec B$ and $\vec C$ so that our $\vec B$ is between $\vec A$ and $\vec C$.) He arrived at negative probabilities, implicitly assuming that (in terms of §\[sec:space\]) there is some solution of the appropriate Extension Problem. In the rest of this section, we explain Schneider’s derivation and then address the implicit assumption.

Imagine that we 