# Problem 4: MAP Again

Consider the situation from Problem 2. Assume that all students in Nafeli’s class are the same way during the exam: for each question, there are two equally likely possibilities, independent of other questions. In the first possibility, they know the answer, in which case they answer the question correctly. In the second possibility, they guess the answer with probability of success 1/3. On this exam, each student in Nafeli’s class is equally likely to belong to one of three categories. i = 1, 2,3: those who know the answer to each question with corresponding probabilities 0,, where Of = 0.3, 02 =0.7, and 03 = 0.95 (independent of other questions). Suppose that a randomly chosen student in Nafeli’s class answers k questions correctly. For each possible value of k, derive the MAP estimate of the category that this student belongs to.

Nafeli is taking a multiple-choice exam with 10 questions and 3 choices per question. For each question, there are two equally likely possibilities, independent of other questions. In the first possibility, she knows the answer, in which case she answers the question correctly. In the second possibility, she guesses the answer with probability of success 1/3.