31. You wake up one morning with severe pain in your kidneys. You've never had pain this severe and persistent in your kidneys before, so you think there is definitely something wrong. Before getting any concrete evidence, you estimate that there is about a 9/10 chance that there is something seriously wrong with you. You go to see the doctor and, after a series of preliminary tests, are told that you might have a rare disorder. Your doctor suggests that you be tested. The test, however, is not conclusive. The test is 90% reliable, meaning that if you in fact have the disorder, the test will detect it nine times out of ten. Alternatively, if you donothave the disorder, the test will correctly identify it nine times out of ten.

The test comes back negative. What is the probability thatyou have the disorder in questiongiven that youreceived a negative test result?

(You have committed the base rate fallacy in taking the probability of a negative test given that you have the disorder for the probability of having the disorder given that you received a negative test result. In other words, the Pr(N|D) for Pr(D|N). Try again. Be sure to include a legend, setup, and calculations. Refer to the lecture notes on Bayes' Rule for solving this type of problem.)

32. You are still unsure whether there is cause for concern, so you ask to be retested. The test, once again, comes back negative.Use your answer from the previous question torevise your prior degree of belief in light of the second negative test result.

(Experience for solving this type of problem)

33. Your friend wants to make a bet with you: You bet $30 and roll a fair, 6-sided die.If you roll a 1,you win $120. But ifyou roll any other number, you lose$30.

a) What is the expected monetary value of taking the bet?

b) Conceptually speaking, what does this mean?

c) What is the fair price of thebet?

(The utility of taking the bet is $120 - $30, since $30 is what you have already paid. Recalculate a) and c). Refer to the lecture notes on Expected Value for calculating the fair price.)