Find , , Lyman Ott Michael Longnecker

4.31 In Example 4.4, we described a new test for determining defects in circuit boards. Com- pute the probability that the test correctly identifies the defects D 1 , D 2 , and D 3 ; that is, compute .

4.32 In Example 4.4, compute the probability that the test incorrectly identifies the defects D

1 , D 2 , and D 3 ; that is, compute . Bus. 4.33 An underwriter of home insurance policies studies the problem of home fires resulting from wood-burning furnaces. Of all homes having such furnaces, 30 own a type 1 furnace, 25 a type 2 furnace, 15 a type 3, and 30 other types. Over 3 years, 5 of type 1 furnaces, 3 of type 2, 2 of type 3, and 4 of other types have resulted in fires. If a fire occurs in a particular home, what is the probability that a type 1 furnace is in the home? Med. 4.34 In a January 15, 1998, article, the New England Journal of Medicine reported on the util- ity of using computerized tomography CT as a diagnostic test for patients with clinically suspected appendicitis. In at least 20 of patients with appendicitis, the correct diagnosis was not made. On the other hand, the appendix was normal in 15 to 40 of patients who underwent emergency appendectomy. A study was designed to determine the prospective effectiveness of using CT as a diagnostic test to improve the treatment of these patients. The study examined 100 consecutive patients suspected of having acute appendicitis who presented to the emergency department or were referred there from a physician’s office. The 100 patients underwent a CT scan, and the surgeon made an assessment of the presence of appendicitis for each of the patients. The final clinical outcomes were determined at surgery and by pathological examination of the appendix after appendectomy or by clinical follow-up at least 2 months after CT scanning. Presence of Appendicitis Radiologic Determination Confirmed C Ruled Out RO Definitely appendicitis DA 50 1 Equivocally appendicitis EA 2 2 Definitely not appendicitis DNA 1 44 The 1996 rate of occurrence of appendicitis was approximately PC ⫽ .00108. a. Find the sensitivity and specificity of the radiological determination of appendicitis. b. Find the probability that a patient truly had appendicitis given that the radiological determination was definite appendicitis DA.

c. Find the probability that a patient truly did not have appendicitis given that the radio-

logical determination was definite appendicitis DA.

d. Find the probability that a patient truly did not have appendicitis given that the radio-

logical determination was definitely not appendicitis DNA. Med. 4.35 Conditional probabilities can be useful in diagnosing disease. Suppose that three different, closely related diseases A 1 , A 2 , and A 3 occur in 25, 15, and 12 of the population. In addi- tion, suppose that any one of three mutually exclusive symptom states B 1 , B 2 , and B 3 may be associated with each of these diseases. Experience shows that the likelihood of having a given symptom state when the disease is present is as shown in the following table. Find the probability of disease A 2 given symptoms B 1 , B 2 , B 3 , and B 4 , respectively. Disease State A i Symptom State B j A 1 A 2 A 3 B 1 .08 .17 .10 B 2 .18 .12 .14 B 3 .06 .07 .08 B 4 no symptoms .68 .64 .68 PB j |A i PD 1 | A 1 , PD 2 | A 2 , and PD 3 | A 3 PD 1 | A 1 , PD 2 | A 2 , and PD 3 | A 3