J lwbelt koch in 1883.) 18 A disease D is present in 2% of the population. Preli

Question

J lwbelt koch in 1883.) 18 A disease D is present in 2% of the population. Preliminary trials of a diagnostic test for the disease show a false positive rate of 2% and a false negativerate of 1%. A physician friend of ours notices that in her practice only about half of the patients who test positive turn out to actually have the disease. She was expecting the number to be much higher. How would you explain to her the reason for her observation? Be as precise as possible but do give an explanation, not just a bunch of calculations

Explanation / Answer

Here, we are given that:

P( disease ) = 0.02, therefore P( no disease ) = 1 - 0.02 = 0.98

P( positive | no disease ) = 0.02

P( negative | disease ) = 0.01

Now using law of total probability, we get:

P( positive ) = P( positive | no disease ) P( no disease ) + P( positive | disease ) P( disease )

P( positive ) = 0.02*0.98 + 0.99*0.02 = 0.0394

Using bayes theorem, we get:

P( disease | positive ) = P( positive | disease ) P( disease ) / P( positive )

P( disease | positive ) = 0.99*0.02 / 0.0394 = 0.5025

Therefore 0.5025 is the probability that a positive test here means that the person really has the disease.

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