  We use an information-theoretic argument due to O'Connell (2000) to prove
that every sufficiently symmetric event concerning a countably infinite family
of independent and identically distributed random variables is deterministic
(i.e., has a probability of either 0 or 1). The i.i.d. condition can be
relaxed. This result encompasses the Hewitt-Savage zero-one law and the
ergodicity of the Bernoulli process, but also applies to other scenarios such
as infinite random graphs and simple renormalization processes.
