Lecture 7

I’ll now sketch a proof. It basically follows by by recalling the definition of the probability of an even conditioned on X2 = a2. We first note that the set of choices for (X 1, X2, 3) given that X2 = a2 is def Sa2 = {(X1, a2, X3) : (X1, a2) ∈ C12 and (a2, X3) ∈ C23} . Conditioning on X2 = a2, we obtain a sample chosen uniformly from Sa2 . Thus, for (a 1, a2, a3) ∈ Sa2 , 1 P [(X1, X2, X3) = (a1, a2, a3)|X2 = a2] = . |Sa2 | Note that Sa2 | = | |{a1 : (a1, a2) ∈ C12}| |{a3 : (a2, a3) ∈ C23}| The claim now follows from observing that