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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?

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With $k$ tries you can get $0,1,2,...,k$ black balls. That are $k+1$ (due to the $0$) possibilities.

It is not a martingale though. With more tries you are more and more certain which hat will give you a black ball more often. The knowledge of past events thus helps you to predict future events.

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That's what I was expecting for the k+1.. because of the 0 possibility. In probability theory, can we say that's because of $\mu(\emptyset)=0$ ? My question ask to show that it is a martingale... at first I also thought of the same...because of the learning process.... – ChuckM Feb 2 '13 at 10:35

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