Alice has a piggy bank inside which she has one biased coin, lets call this coin A. It is known that coin A shows heads 2 3 of the times it is tossed up. Bob has an unbiased coin, lets call it coin B (which outwardly looks identical to coin A). He unknowingly drops the coin inside the piggy bank. Now the piggy bank has two coins and Alice wants to find her coin in the piggy bank, i.e., identify which coin is coin A. She decides to perform the following experiment. She picks out a coin from the piggy bank at random and flips it twice independently and records the outcomes as X1, X2, where Xi = 1, i = 1, 2 if the ith toss yielded a heads and Xi = 0 otherwise. Let = {1, 2} be the random variable associated with which coin Alice picks up. Given the results of the coin tosses, Alice has to find out if she picked coin A or coin B. a. Write the conditional probabilities of Xi , i = 1, 2 given . b. Alice asks for your help for coming up with a decision rule D(X1, X2) {1, 2} for deciding which coin she picked up, i.e., determining whether = 1 or 2. What is the decision rule you will suggest to minimize the error probability? You need to give a precise answer. Hint: Recall that the rule minimizing the error probability is the maximum a-posteriori decision rule, which picks max P[choice | observation]. 1