= R(3)+R(7) = \frac18 + \frac18(1-\sqrt2) = \frac18(2-\sqrt2)\end{aligned}$$

Hardy’s Gedankenexperiment: Contextuality without negativity {#sec:hardy}
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The previous section may give one the idea that contextuality always leads to negative probabilities. In this section, building on Lucien Hardy’s article [@Hardy] and also influenced by David Mermin’s article [@Mermin], we illustrate that this is not so.

Two one-qubit particles emerge from a common source heading for two far apart detectors. Aside from the passage of the particles from the source to the detectors, there are no connections between the source and either detector or between the two detectors. The following picture is borrowed from [@Mermin] (and slightly modified).

Each of the detectors is randomly set, ahead of time, to one of two modes, indicated by “1” and “2” in the picture. Four cases arise, two possible settings on each of the two detectors. When a particle arrives at a detector, that detector performs a measurement and exhibits the result. In mode 1, observable $Z$ is measured. Its value is $+1$ in state 0 and $-1$ in state 1. In mode 2, observable $X$ is measured. Its value is $+1$ in state ${\ensuremath{|+\rangle}} = \frac1{\sqrt2}({\ensuremath{|0\rangle}} + {\ensuremath{|1\rangle}})$ and $-1$ in state ${\ensuremath{|-\rangle}} = 