Past exam of the quantum information course of the University of Cambridge / 2015 / Paper 1 / 2 / a / Solution

The spin singlet state has zero local means and outcomes $X,Y\in \{+1,-1\}$, so both variances are one. Its Pearson correlation coefficient is therefore $C=\mathbb{E}[XY]$. Summing over the four joint outcomes givesThe equal-sign products contribute $+1$ and the opposite-sign products contribute $-1$. In a finite sample the corresponding relative frequencies estimate these probabilities; the formula is not an assertion that a finite random sample has exactly zero empirical means.