  Probabilistic cellular automata with deterministic updating are quantum
systems. We employ the quantum formalism for an investigation of random
probabilistic cellular automata, which start with a probability distribution
over initial configurations. The properties of the deterministic updating are
randomly distributed over space and time. We are interested in a possible
continuum limit for a very large number of cells. As an example we consider
bits with two colors, moving to the left or right on a linear chain. At
randomly distributed scattering points, they change direction and color. A
numerical simulation reveals the typical features of quantum systems. We find
particular initial probability distributions which reemerge periodically after
a certain number of time steps, as produced by the periodic evolution of energy
eigenstates in quantum mechanics. Using a description in terms of wave
functions allows to introduce statistical observables for momentum and energy.
They characterize the probabilistic information without taking definite values
for a given bit configuration, with a conceptual status similar to temperature
in classical statistical thermal equilibrium. Conservation of energy and
momentum are essential ingredients for the understanding of the evolution of
our stochastic probabilistic automata. This evolution resembles in some aspects
a single Dirac fermion in two dimensions with a random potential.
