How should we model an observer within quantum mechanics or quantum field theory? How can classical physics emerge from a quantum model, and why should classical probability be useful? How can we model a selective measurement entirely within a closed quantum system? This paper sketches a new physical theory of probability based on an attempt to model classical information within a purely quantum system. We model classical information using a version of Zurek's theory of Quantum Darwinism, with emphasis on quantum information encoded using projection operators localised in spacetime. This version of Quantum Darwinism is compatible with quantum field theory, and does not require any artificial division of a quantum system into subsystems. The main innovation is our attempt to provide a physical explanation of probability. Decoherence is the physical mechanism behind Quantum Darwinism or the `branching of quantum worlds'. Assuming a type of perfect decoherence we construct a conventional probabilistic model for classical information. This, however, is not our theory of natural probability, and does not quite demonstrate the validity of Bayesian reasoning. Instead, our theory of natural probability arises from careful consideration of errors in decoherence: roughly speaking, we don't observe low probability events because they are swamped by quantum noise.