Dr. Wellby\'s patient exhibits symptoms associated with acute neural toxemia (an

Question

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 25% of those who exhibit the symptoms actually have acute neural toxemia (ANT). A diagnostic test correctly diagnoses the presence of ANT in 88% 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's good health will return under this plan. (Assume no patients experience side effects during treatment. Round your answer to four decimal places.)


(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's good health will return under this plan. (Assume no patients experience side effects during treatment. Round your answer to four decimal places.)


(c) Find the probability that the patient's good health will return if he undergoes treatment regardless of the test's outcome. (Round your answer to four decimal places.)


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

Yes/No   

Explanation / Answer

(a) The probability that a patient's good health will return under this plan is, P( the patient does not have ANT but the symptoms and tested negatively) + P( Patient has ANT tested positively and treated ) + P( Patient has ANT tested negatively and survived ) = 0.75*0.91 + 0.25*0.88 + 0.25*(1-0.88)*(1-0.9) = 0.6825 + 0.22 + 0.003 = 0.9055 .

(b) P( patient's ood health returns ) = P( Patient does not have ANT ) + P( Patient has ANT and survived ) = 0.75 + 0.25*(1-0.9) = 0.75 + 0.025 = 0.775 .

(c) P( Patient's good health returns when he undergoes treatment regardless the test's outcome) = P( Patient has ANT undergoes treatment and no side effect ) + P( Patient does not have ANT undergoes treatment and cured ) = 0.25*(1-0.02) + 0.75 = 0.245 + 0.75 = 0.995

(d) Yes he would recommend for testing as the cases when test performed the probabilities that a patient will regain good health are high as 0.9055 and 0.995 .

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