(Optional) A teacher wants to find out how confident the students are about their own abilities. He.

(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

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