3. Three coins. There are three identically looking coins. For coin 1, the probability of heads is 0.3 and the probability of tails is 0.7; for coin 2, the probability of heads is 0.7 and the probability of tails is 0.3; for coin 3, the probability of heads is 0.5 and the probability of tails is 0.5. (a) Suppose Alice was given one of the three coins uniform at random. She flips the given coin twice (independent of each other) and got heads on the first flip and tails on the second flip. Based on this outcome, Alice wants to estimate which coin was given to her. Find the optimal estimate that minimizes the probability of error. (b) Suppose Alice was given coin 1 with probability 0.6 and coin 2 with probability 0.4 (which implies she was given coin 3 with probability 0). She wants to estimate which coin was given to her from the outcome of flipping the given coin once. Find the MAP estimator given the outcome of a single flip. Find the corresponding (overall) probability of error.

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