  Spin is a fundamental and distinctive property of the electron, having
far-reaching consequences in wide areas of physics. Yet, further to its
association with an angular momentum, the physics underpinning its formal
treatment remains obscure. In this work we propose to advance in disclosing the
meaning behind the formalism, by first recalling some basic facts about the
one-particle spin operator. Consistently informed by and in line with the
quantum formalism, we then proceed to analyse in detail the spin projection
operator correlation function $C_Q(\boldsymbol{a}, \boldsymbol{b}) =
\langle(\hat{\sigma}\cdot\boldsymbol{a})(\hat{\sigma}\cdot\boldsymbol{b})\rangle$
for the bipartite singlet state, and show it to be amenable to an unequivocal
probabilistic reading. In particular, the calculation of $C_Q(\boldsymbol{a},
\boldsymbol{b})$ entails a partitioning of the probability space, which is
dependent on the directions $(\boldsymbol{a}, \boldsymbol{b})$. The derivation
of the CHSH- or other Bell-type inequalities, on the other hand, does not
consider such partitioning. This observation puts into question the
applicability of Bell-type inequalities to the bipartite singlet spin state.
