  The theory of $F$-manifolds, and more generally, manifolds endowed with
commutative and associative multiplication of their tangent fields, was
discovered and formalised in various models of quantum field theory involving
algebraic and analytic geometry, at least since 1990's.
  The focus of this paper consists in the demonstration that various spaces of
probability distributions defined and studied at least since 1960's also carry
natural structures of $F$-manifolds.
  This fact remained somewhat hidden in various domains of the vast territory
of models of information storing and transmission that are briefly surveyed
here.
