I need help with this problem Problem 9. Consider a game in which balls are divided among three boxes. First, I choose a box from among the three boxes, and then I choose a ball at random from that particular box. Box 1 contains 1 white and 1 black ball; Box 2 contains 2 white and 1 black ball; Box 3 contains 3 white and 1 black ball. After you see what color ball I pick, you guess which Box I chose from, and the game is played over and over. Your objective is to guess the box correctly as often as possible.a) Suppose you know that I pick a box at random with probability (1/3, 1/3, 1/3). Your strategy is to guess the box with highest posterior probability given the observed color. What fraction of the time are you likely to be correct?b) Can you do better given any other strategy?c) Suppose you use the guessing strategy from (a), but I was in fact randomizing with probabilities (1/2, 1/4, 1/4) instead. Now how would your strategy perform?d) Suppose you knew I was randomizing with either set of prior probabilities (but the same one every time). How would you guess which one, and what would you do?
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Problem 9. Consider a game in which balls are divided among three boxes. First, I choose a box from among the three boxes, and then I choose a ball at random from that particular box. Box 1 contains 1 white and 1 black ball; Box 2 contains 2 white and 1 black ball; Box 3 contains 3 white and 1 black ball. After you see what color ball I pick, you guess which Box I chose from, and the game is played over and over. Your objective is to guess the box correctly as often as possible.
a) Suppose you know that I pick a box at random with probability (1/3, 1/3, 1/3). Your strategy is to guess the box with highest posterior probability given the observed color. What fraction of the time are you likely to be correct?
b) Can you do better given any other strategy?
c) Suppose you use the guessing strategy from (a), but I was in fact randomizing with probabilities (1/2, 1/4, 1/4) instead. Now how would your strategy perform?
d) Suppose you knew I was randomizing with either set of prior probabilities (but the same one every time). How would you guess which one, and what would you do?
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