Chapter 5.4, Problem 61PB

DNA evidence compelling? DNA evidence can be extracted from biological traces such as blood, hair, and saliva. “DNA fingerprinting” is increasingly used in the courtroom as well as in paternity testing. Given that a person is innocent, suppose that the probability of his or her DNA matching that found at the crime scene is only 0.000001, one in a million. Further, given that a person is guilty, suppose that the probability of his or her DNA matching that found at the crime scene is 0.99. Jane Doe’s DNA matches that found at the crime scene.

1. a. Find the probability that Jane Doe is actually innocent, if absolutely her probability of innocence is 0.50. Interpret this probability. Show your solution by introducing notation for events, specifying probabilities that are given, and using a tree diagram to find your answer.
2. b. Repeat part a if the unconditional probability of innocence is 0.99. Compare results.
3. c. Explain why it is very important for a defense lawyer to explain the difference between P(DNA match | person innocent) and P(person innocent | DNA match).

5.57 Mammogram diagnostics Breast cancer is the most common form of cancer in women, affecting about 10% of women at some time in their lives. There is about a 1% chance of having breast cancer at a given time (that is, P(S) = 0.01 for the state of having breast cancer at a given time). The chance of breast cancer increases as a woman ages, and the American Cancer Society recommends an annual mammogram after age 40 to test for its presence. Of the women who undergo mammograms at any given time, about 1% are typically estimated to actually have breast cancer. The likelihood of a false test result varies according to the breast density and the radiologist’s level of experience. For use of the mammogram to detect breast cancer, typical values reported are sensitivity = 0.86 and specificity = 0.88.

1. a. Construct a tree diagram in which the first set of branches shows whether a woman has breast cancer and the second set of branches shows the mammogram result. At the end of the final set of branches, show that P(S and POS) = 0.01 × 0.86 = 0.0086 and report the other intersection probabilities also.
2. b. Restricting your attention to the two paths that have a positive test result, show that P(POS) = 0.1274.
3. c. Of the women who receive a positive mammogram result, what proportion actually have breast cancer?
4. d. The following tree diagram illustrates how P(S|POS) can be so small, using a typical group of 100 women who have a mammogram. Explain how to get the frequencies shown on the branches and why this suggests that P(S|POS) is only about 0.08.

Typical results of mammograms for 100 women