5.110 lity. What additional 5.105 Medical risks. You have torn a tendon and are

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5.110
lity. What additional 5.105 Medical risks. You have torn a tendon and are facing surgery to repair it. The surgeon explains the risks to you: infection occurs in 3% of such operations, the repair fails in 14%, and both infection and failure occur together in 1%. What percent of these operations succeed and are free from infection? antibodies to HIV, the virus that causes AIDS. Antibodies indicate the presence of the virus. The test is quite accurate but is not always comect. Here are approximate probabilities of positive and negative ElA outcomes when the blood tested does and does not actually contain antibodies to HIV Working. In the language of government statistics, you are "in the labor force" if you are available for work and either working or actively seeking work. The unemployment rate is the proportion of the labor force (not of the entire population) who are unemployed. Here are data from the Current Population Survey for the civilian population aged 25 years and over. The table entries are counts in thousands of people. Exercises 5.106 to 5.108 concern these data. Test Result Antibodies present 0.9985 0.0015 Antibodies absent 0.006 0.994 Suppose that 1% of a large population carries antibodies to HIV in their blood. (a) Draw a tree diagram for selecting a person from this population (outcomes: antibodies present or absent) and for testing his or her blood (outcomes: ElA positive or negative). Total In labor Highest education population force Employed (b) What is the probability that the EIA is positive for a randomly chosen person from this population? Did not finish 27,325 12,073 11,139 (c) What is the probability that a person has the antibody, given that the ElA test is positive? (Comment: This exercise illustrates a fact that is important when considering proposals for widespread testing for HIV, illegal drugs, or agents of biological warfare: if the condition being tested is uncommon in the population, many positives will be false-positives.) High school but 57,221 36,855 35,137 Less than bachelor's 45,471 33,331 31,975 degree College graduate 47,371 37,281 36,259 5.110) Testing for HIV, continued. The previous exercise Unemployment rates. Find the unemployment rate for people with each level of education. How does the unemployment rate change with education? Explain carefully why your results show that level of education and being employed are not independent. gives data on the results of ElA tests for the presence of antibodies to HIV. Repeat part (c) of that exercise for two different populations: 5.106 (a) Blood donors are prescreened for HIV risk factors, so perhaps only 0.1% (0001) of this population carries HIV antibodies. 5.107 Education and work. (a) What is the probability that a randomly chosen person 25 years of age or older is in the labor force? (b) Clients of a drug rehab clinic are a high-risk group, so perhaps 10% of this population carries HIV antibodies. (b) If you know that the person chosen is a college graduate, what is the conditional probability that he or she is in the labor force? (c) What general lesson do your calculations illustrate? (c) Are the events "in the labor force" and "college graduate" independent? How do you know? The geometric distributions. You are tossing a balanced die that has probability 1/6 of coming up 5.111 1 on each toss. Tosses are independent. We are interested in how long we must wait to get the first l Education and work, continued. You know that a person is employed. What is the conditional probability that he or she is a college graduate? You 5.108 (a) The probability of a 1 on the first toss is 1/6. What is the probability that the first toss is not a l an the second toss is a 1? that a second person is a college graduate. What is the conditional probability that he or she is (b) What is the probability that the first two tosses a Testing for HIV. Enzyme immunoassay (EIA) tests are used to screen blood not Is and the third toss is a 1? This is the probabili that the first 1 occurs on the third toss, 5.109 specimens for the presence of

Explanation / Answer

Bayes theorem - P(A | B) = P(A and B)/P(B)

P(HIV anitbody | EIA test is positive) = P(HIV antibody and positive EIA test)/P(positive EIA test)

a) When 0.1% carries antibodies, P(HIV antibody | EIA test is positive)

= (0.001x0.9985)/(0.001x0.9985+0.999x0.006)

= 0.1428

b) When 10% carries antibodies, P(HIV antibody | EIA test is positive)

= (0.1x0.9985)/(0.1x0.9985+0.9x0.006)

= 0.9487

c) The calculation illustrates that with increase in the percentage of people carrying the antibodies, there is more chance that a positive test will indicate that a person is actually a carrier of antibody. So, the accuracy of the test is dependant on the prevalence.

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