The Pap smear is the standard test for cervical cancer. The false-positive rate

Question

The Pap smear is the standard test for cervical cancer. The false-positive rate is .583; the false-negative rate is .180. Family history and age are factors that must be considered when assigning a probability of cervical cancer. Suppose that, after obtaining a medical history, a physician determines that 2% of women of his patient's age and with similar family histories have cervical cancer. Determine the effects a positive Pap smear test has on the probability that the patient has cervical cancer. (to 4 decimals).

Note: false-positive means that the test shows cancer even though the patient doesn't have cancer. Similarly, false-negative means that the test shows NO cancer, even though the patient has it.

Explanation / Answer

Here, we are given that: false-positive rate is .583; the false-negative rate is .180, therefore:

P( positive | no cancer ) = 0.583
P( negative | cancer ) = 0.180, therefore P( positive | cancer ) = 1 - 0.18 = 0.82

Also, we are given that P( cancer ) = 0.02, therefore P( no cancer ) = 1 - P( cancer ) = 0.98

Using law of total probability, we get:

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

P( positive ) = 0.82*0.02 + 0.583*0.98 = 0.58774

Now using bayes theorem, we get:

P( cancer | positive ) = P( positive | cancer )P( cancer ) / P( positive )

P( cancer | positive ) = 0.82*0.02 / 0.58774

P( cancer | positive ) = 0.0279

Therefore 0.0279 is the required probability here.

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