Question

In: Statistics and Probability

If a patient has a disease there is a 93.8% chance that the test gives a...

If a patient has a disease there is a 93.8% chance that the test gives a correct positive response and if a patient does not have this disease, there is a 95.6% chance that the test gives a correct negative response. About 20 out of every 1000 members have the disease.

i) A member has the disease given that they have a positive test response

ii) A member has the disease given that they have a negative test response

Expert Solution

P(having the disease) = 20/1000 = 0.02

P(positive test response | having the disease) = 0.938

P(negative test response | not having the disease) = 0.956

a) P(positive test response | not having the disease) = 1 - P(negative test response | not having the disease) = 1 - 0.956 = 0.044

P(positive test response) = P(positive test response | having the disease) * P(having the disease) + P(negative test response | not having the disease) * P(not having the disease)

= 0.938 * 0.02 + 0.044 * (1 - 0.02)

= 0.06188

P(has the disease | positive test response) = P(positive test response | having the disease) * P(having the disease) / P(positive test response)

= 0.938 * 0.02 / 0.06188

= 0.3032

b) P(negative test response) = 1 - P(positive test response) = 1 - 0.06188 = 0.93812

P(negative test response | having the disease) = 1 - P(positive test response | having the disease) = 1 - 0.938 = 0.062

P(has the disease | negative test response) = P(negative test response | having the disease) * P(having the disease) / P(negative test response)

= 0.062 * 0.02 / 0.93812

= 0.0013

orchestra answered 1 year ago

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