Conditioning and Martingales

Let (Ω,F, P) be a probability space and A, B ∈F events such that P(B) > 0. In this section we seek to formalize the intuitive content of the phrase “probability of A given (or, equivalently, “conditioned on”) B”. The words “given B ” imply prior knowledge of the fact that the sample point ω lies in B. It is then natural to consider the reduced probability space (Ω, F_b, P_b) with Ω_B = B,F_B = A∩B A ∈ F (the “trace σ-field”), P_B(C) = P(C)/P(B) for C ∈ T_B, and define the probability of A given B, denoted by P(A/B), as P_B(A ∩ B) = P(AB)/P(B). The reader may convince himself of the reasonableness of this procedure by evaluating, say, the probability of two consecutive heads in four tosses of a fair coin given the knowledge that the number of heads was even, using the principle of insufficient reason.