  On the affine space containing the space $\mathcal{S}$ of quantum states of
finite-dimensional systems there are contravariant tensor fields by means of
which it is possible to define Hamiltonian and gradient vector fields encoding
relevant geometrical properties of $\mathcal{S}$. Guided by Dirac's analogy
principle, we will use them as inspiration to define contravariant tensor
fields, Hamiltonian and gradient vector fields on the affine space containing
the space of fair probability distributions on a finite sample space and
analyse their geometrical properties.
  Most of our considerations will be dealt with for the simple example of a
three-level system.
