Dr Wellbys patient exhibits symptoms associated with acute n

Dr. Wellby\'s patient exhibits symptoms associated with acute neural toxemia (an imaginary disease), but the symptoms can have other, innocuous causes. Studies show that only 26% of those who exhibit the symptoms actually have acute neural toxemia (ANT). A diagnostic test correctly diagnoses the presence of ANT in 87% of the persons who have it and correctly diagnoses its absence in 91% of the persons who do not have it. ANT can be successfully treated, but the treatment causes side effects in 2% of the patients. If left untreated, 90% of those with ANT die; the rest recover fully. Dr. Wellby is considering ordering a diagnostic test for her patient and treating the patient if test results are positive, but she is concerned about the treatment\'s side effects.

(a) Dr. Wellby could choose to test her patient and administer the treatment if the results are positive. Find the probability that her patient will live under this plan. (Round the answer to one decimal place.)

(b) Dr. Wellby could choose to avoid the treatment\'s side effects by not administering the treatment. (This also implies not testing the patient.) Find the probability that her patient will live under this plan. (Round the answer to one decimal place.)


(c) Find the probability that the patient\'s good health (no side effects) will return if he undergoes treatment regardless of the test\'s outcome. (Round the answer to one decimal place.)


(d) On the basis of the probabilities, should Dr. Wellby order the test and treat the patient if test results are positive?

Solution

(a) the probability that her patient will live under this plan = .26*.87 = 0.2262 = .2

(b) the probability that her patient will live under this plan = .1*.87 + .74 = 0.827 = .8

(c) the probability that the patient\'s good health (no side effects) will return if he undergoes treatment regardless of the test\'s outcome = .87*.98 = 0.8526 = .8

(d) yes she should order the test and treat the patient