4. Suppose a test for a serious disease is successful in detecting the disease i

Question

4. Suppose a test for a serious disease is successful in detecting the disease in 95% of all persons infected but that it incorrectly diagnoses 2% of all healthy people as having the serious disease. Suppose also that it incorrectly diagnoses 10% of all people having another minor disease as having the serious disease. It is known that 5% of the population has the serious disease, 5% has the minor disease, and 90% ofpopulation is healthy. Find the probability that a person selected at random really has serious disease if the test indicates that he or she does. 96

Explanation / Answer

Here we are given that 5% of the people have serious disease. Therefore, we have here:

P( serious) = 0.05

Then we are given that 5% has minor disease. Therefore P( minor ) = 0.05 and therefore P( healthy ) = 0.9

Then we are given that sensitivity of the test is 95%. This means that:

P( positive | serious ) = 0.95

Also it is given that:

P( positive | healthy ) = 0.02 and P( positive | minor ) = 0.1

Now using the law of total probability, the probability that the test is positive is computed as:

P( positive ) = P( positive | serious )P(serious) + P( positive | minor ) P(minor ) + P( positive | healthy ) P( healthy)

P( positive ) = 0.95*0.05 + 0.1*0.05 + 0.02*0.9 = 0.0705

Now by bayes theorem, given that the test is positive, probability that the person selected has a serious disease is computed as:

P( serious | positive ) = P( positive | serious )P(serious) / P( positive )= 0.95*0.05 / 0.0705 = 0.6738

Therefore 0.6738 is the required probability here.

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