In this discussion, we indicate possibilities for (homological and non-homological) linearization of basic notions of the probability theory and also for replacing the real numbers as values of probabilities by objects of suitable combinatorial categories. The success of the probability theory decisively, albeit often invisibly, depends on symmetries of systems this theory applies to. For instance: • The symmetry group of a single round of gambling with three dice has order 288 = 6 × 6 × 8: it

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... s a semidirect product of the permutation group S 3 of order 6 and the symmetry group of the 3d cube, that is, in turn, is a semidirect product of S 3 and {±1} 3 . • The Bernoulli spaces (gp, p 1−p ) Z , 0 < p < 1, of (g, p)-sequences indexed by integers z ∈ Z = {⋯, −2, −1, 0, 1, 2, ⋯} are acted upon by a semidirect product of the infinite permutation group S ∞=Z ⊃ Z = {⋯, −2, −1, 0, 1, 2, ⋯} and the (compact) group {±1} Z = { g ↔ p} Z , with the role of the latter being essential even for p ≠ 1 2 where the probability measure is not preserved. • The system of identical point-particles • i in the Euclidean 3-space R 3 , that are indexed by a countable set I ∋ i, is acted upon by the isometry group of R 3 times the infinite permutation group S ∞=I .