I have some difficulties at solving a traditional problem where we have two hats, hat A and B, where there are black and white balls in each hats but the experimenter does not know the proportion of black balls in hat A and proportion for hat B. Let with . The proportion of hat A is . The experimenter draws randomly some balls (with replacement) to determine which of the hats he is drawing from. After each draws, the experimenter updates his beliefs using Bayes' rule. Denote where the experimenter posterior probability after k balls have been drawn.

My questions are:

(1) why is a random variable that can take k+1 values? I don't see where the +1 comes from... that's probably because I am not sure how to define the probability space.

(2) is this a martingale due to the replacement of a ball back into the hat after each draws?