The probability that a person has a certaindisease is 0.03. Medical diagnostic t

Question

The probability that a person has a certaindisease is 0.03. Medical diagnostic tests are available todetermine whether the person actually has the disease. If thedisease is actually present, the probability that the medicaldiagnostic test will give a positive result (indicating that thedisease is present) is 0.9. If the disease is not actually present,the probability of a positive test result (indicating that thedisease is present) is 0.02. Suppose that the medical diagnostictest has given a positive result (indicating that the disease ispresent).

a) What is the probability that the disease isactually present (given a positive test result)?

b) What is the probability of a positive testresult?

Explanation / Answer

Step 1: Define terms and variables.

Let Pos be the event that someone is tested positive.

Let Neg be the event that someone is tested negative.

Let D and not D be the events of having and not having thedisease.

P(D) = 0.03

P(Pos | D) = 0.9

P(Pos | not D) = 0.02

Step 2: Find the probability of tested positive.

   = P(D) * P(Pos | D) + P(Not D) * P(Pos | notD)

    = 0.03 * 0.90 + 0.97 * 0.02 = 0.020194

Step 3: Find the probability of disease given testedpositive.

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