is proportional to the angle subtended at the centre.) The formula () above has not been used explicitly – it is hardly relevant when, as here, the situation is fully deterministic. The problem could be reformulated in terms of the functions in the formula, with ρ constant and the probability functions step functions. The principle behind () has in fact been used, but purely intuitively. Thus the local hidden-variable prediction for the probability of coincidence is proportional to the angle (b − a) between the detector settings. The quantum correlation is defined to be the expectation value of the sum of the individual outcomes, and this is where P++ is the probability of a "+" outcome on both sides, P+− that of a "+" on side A, a "−" on side B, etc. Since each individual term varies linearly with the difference (b − a), so does their sum. The result is shown in the figure. Optical Bell tests In almost all real applications of Bell's inequalities, the particles used have been photons. It is not necessarily assumed that the photons are particle-like. They may be just short pulses of classical light. It is not assumed that every single one is detected. Instead the hidden variable set at the source is taken to determine only the probability of a given outcome, the actual individual outcomes being partly determined by other hidden variables local to the analyser and detector. It is assumed that these other hidden variables are independent on the two sides of the experiment. In this stochastic model, in contrast to the above deterministic case, we do need equation () to find the local-realist prediction for coincidences. It is necessary first to make some assumption regarding the functions and , the usual one being that these are both cosine squares, in line with Malus' law. Assuming the hidden variable to be polarisation direction (parallel on the two sides in real applications, not orthogonal), equation () becomes where . The predicted quantum correlation can be derived from this and is shown in the figure. In optical tests, incidentally, it is not certain that the quantum correlation is well-defined. Under a classical model of light, a single photon can go partly into the "+" channel, partly into the "−" one, resulting in the possibility of simultaneous detections in both. Though experiments such as by Grangier et al. have shown that this