(Optional) A teacher wants to find out how confident the students are about their own abilities. He proposes the following scheme: “After you answer this question, state your estimate of the...

(Optional) A teacher wants to find out how confident the students are about their own abilities. He proposes the following scheme: “After you answer this question, state your estimate of the probability that you are right. I will then check your answer to the question. Suppose you have given the probability estimate x. If your answer is actually correct, your grade will be log(x). If incorrect, it will be log(1 2 x).” Show that this scheme will elicit the students’ own truthful estimates—that is, if the truth is p, show that a student’s stated estimate x = p