Find the probability that a customer chosen at random will pay exactly one month in full.

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 4.6 Variables: Discrete and Continuous 4.36 Classify each of the following random variables as either continuous or discrete: a. The lifelength of the battery in a smoke alarm b. The number of rain delays during a baseball game played in Seattle during the month of March

c. The thickness of ice 20 feet from the shoreline in Lake Superior during a random day

in December d. The amount of medication prescribed to a patient having high blood pressure e. The speed at which a major league baseball player throws a baseball f. The amount of water spread on a lawn during a random July day in Kansas 4.37 A state consumer bureau is investigating the impact of the state’s new “lemon law” by inspecting new cars on randomly selected car dealerships. The inspectors were looking for defects on the exterior of the cars dents, misfitting doors, uneven painting, etc.. The inspectors record the number of defects per car. Is the number of defects on a randomly selected car a discrete or continuous random variable? Explain your answer. 4.38 The running of red lights by drivers is a serious problem in many cities. A police officer is stationed near a major intersection to observe the traffic for several days.

a. Is the number of cars running a red light during a given light cycle a discrete or

continuous random variable?

b. Is the time between the light turning red and the last car passing through the inter-

section a discrete or continuous random variable? c. Are the brands of cars running a red light a discrete or continuous random variable? 4.39 Every semester, students are given a questionnaire to evaluate their instructor’s teaching. The question that is of greatest interest to administrators is, “Do you agree with the following statement: ‘overall the instructor was a good teacher.’” The possible responses are Strongly agree, Agree, No opinion, Disagree, and Strongly disagree.

a. Are the number of students in class responding Strongly agree a continuous or discrete

random variable?

b. Are the percent of students in class responding Strongly agree a continuous or discrete

random variable? 4.7 Probability Distributions for Discrete Random Variables Bus. 4.40 An appliance store has the following probabilities for y, the number of major appliances sold on a given day: y Py .100 1 .150 2 .250 3 .140 4 .090 5 .080 6 .060 7 .050 8 .040 9 .025 10 .015

a. Construct a graph of Py. b. Find .

c. Find . d. Find .

P5 ⱕ y ⬍ 9 Py ⱖ 8 Py ⱕ 3