Question

A test for a particular illness that affects 3% of the general population has the following properties: The test will show a positive in 97% of cases where the patient has the illness and a negative in 3% of such cases. The test will show a false positive in 6% of patients who do not have the illness and a negative result for 94% of patients who do not have the illness. Note that a positive result means that the test indicates that the patient has the illness. A negative result means that the test indicates that the patient doesn't have the illness. A patient has just been tested and shows a positive result. Assuming all other things being equal, and based on the above information, find the probability that the patient does not have the illness. Give your answer as a decimal to 2 decimal places.

Probability that patient does not have illness =

Answer 1

Here, we are given that:

P( illness ) = 0.03
P( + | illness ) = 0.97
P( - | illness ) = 0.03
P( + | no illness ) = 0.06
P( - | no illness ) = 0.94

Using law of total probability, we get:

P( + ) = P( + | illness )P( illness ) + P( + | no illness )P( no illness )

P( + ) = 0.97*0.03 + 0.06*(1 - 0.03) = 0.0873

P( no illness | + ) =  P( + | no illness )P( no illness ) / P( + )

P( no illness | + ) =  0.06*(1 - 0.03) / 0.0873

P( no illness | + ) = 2/3 = 0.6667

Therefore 0.67 is the required probability here.