The probability that a person has a certain disease is 0.03. Medical diagnostic

Question

The probability that a person has a certain disease is 0.03. Medical diagnostic tests are available to determine whether the person actually has the disease. If the disease is actually present, the probability that the medical diagnostic test will give a positive result (indicating that the disease is present) is 0.88. If the disease is not actually present, the probability of a positive test result (indicating that the disease is present) is 0.02.

a. If the medical diagnostic test has given a positive result (indicating that the disease is present), what is the probability that the disease is actually present?

b. If the medical diagnostic test has given a negative result (indicating that the disease is not present), what is the probability that the disease is not present?

a.) The probability is ?   (Round to three decimal places as needed.)

b.) To determine the probability that if the medical diagnostic test has not given a positive result, the disease is not actually present, use Bayes' theorem again.

Explanation / Answer

a)

P(positive result)=P(disease present and test positive)+P(disease not present and test positive)

=0.03*0.88+(1-0.03)*0.02=0.0458

hence P(disease present given positive result) =P(disease present and test positive)/P(positive result)

=0.03*0.88/0.0458=0.5764

b)

P(test negative)=1-P(test positive) =1-0.0458=0.9542

hence P(disease not present given test negative)

=P(disease not present and test negative)/P(test negative)=(1-0.03)*(1-0.02)/0.9542 =0.9962

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