Cellular automata display an extraordinary range of behavior, ranging from very simple to apparently chaotic, with many cases in between. Perhaps the most interesting rules are those that yield multiple behavior types from different initial conditions - this is common even for one-dimensional rules started from finitely-supported seeds. If a rule yields chaos from some initial conditions, it is tempting to conclude by analogy with the second law of thermodynamics that chaos should be prevalent from almost all initial conditions. For a certain natural class of rules, we prove that the opposite holds: typical (i.e. random) initial seeds self-organize into predictable (but non-trivial) evolution, while exceptional seeds generate more complicated behavior, including apparent chaos. The key technique is application of percolation arguments to the highly non-independent setting of space-time configurations of cellular automata.