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 $a$ black balls in hat A and proportion $b$ for hat B. Let $0 \le a, b \le 1$ with $a \ne b$. The proportion of hat A is $p_0 \in (0,1)$. The experimenter draws randomly some balls (with replacement) to determine which of the hats he is drawing from. After each $k$ draws, the experimenter updates his beliefs using Bayes' rule. Denote $p_k$ where $k=1,2,...,$ the experimenter posterior probability after k balls have been drawn.
My questions are: (1) why is $p_k$ 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?