1. To review false-positives and false-negatives and the notation we use for conditional probabilities, go to the cancer example discussed on pp. 131-33 of the Plous book, under the heading "Confusion of the Inverse". Note that it represents information both verbally and with probability notation.

a. In the first paragraph it says: "A mammogram is an X-ray test that accurately classifies roughly 80 percent of malignant tumors and 90 percent of benign tumors." Write these two facts using the notation of conditional probability.

b. Use the information above to figure out what the rates of false positives and false negatives on this test are and represent each using the notation of conditional probability.

c. In this example the doctor originally thought the woman was very unlikely to have cancer. After the test comes back positive, what is the probability that the doctor should now assign to the woman's having cancer according to this section? Write it as a conditional probability and say what it's numerical value is. Is this value the same as any of the probabilities you wrote down in the answers to (a) and (b) above? Would its value be larger or smaller or the same if the rate of false positives were to decrease?

Attached is a shortened version of a recent newstory about the accuracy of identifications based on fingerprints. If a fingerprint expert says there is a positive match between two sets of fingerprints but they in fact come from two different people, we will call such an error a "false positive". (What would a case of a "false negative" be?)

Inserted into the linked story at various points are bold letters which refer to the following questions:

i) Which kind of error is being referred to in this passage?

“On a 1995 proficiency test of 156 examiners conducted with the approval of the International Association of Identification, the profession's certifying organization, one in five examiners made at least one misidentification -- linking a mock crime-scene print to the wrong person. Fingerprint experts point out that the error rate was lower on subsequent tests.”

j) Which kind in this passage?

“In that case, both the prosecution and defense fingerprint experts mismatched a print to the defendant.”

k) Briefly describe two kind of medical misdiagnoses. Typically, which kind of error causes more harm to the patient. Briefly explain.

l) What kind of error is under discussion here?

“"If they want to go in and testify, 'I think it's his print and 1 percent of the time I'm wrong,' then that would be more reasonable," Mr. Cole said.”

m) What kind of error is under discussion here? What is the minimum value of the rate of error according to this study?

“But 8 of the 34 laboratories that responded were unable to find a match for at least one of the two latent prints.”

n) In the typical case, which rate of error is a defense lawyer likely to be more interested in and employ in making the case for the defendant, the rate of false positives or the rate of false negatives? Very briefly explain your answer.

o) Fingerprints are a standard item used by detectives, who try to solve cases, but don't have to prosecute them. Which error rate will be of primary concern to them?

p) Suppose you are a suspect who has just been brought in and are being fingerprinted. You have heard that prints were found at the scene of the crime. As you are being fingerprinted would you be more or less nervous if you believed that fingerprinting is subject to a high rate of false positives? Answer the same question for a high rate of false negatives. Does your nervousness depend on whether or not you are guilty? Explain any other assumptions underlying your answers.

3. In class you will get a two-sheet self-test and data about weather forecasters. Fill out the self-test (without peeking!) and calibrate your answers using a graph. (See below.) Did your results indicate over- or under-confidence?

4. Answer the question about the weather forecasters using an appropriate graph.