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Bayes’ Law

Using Bayes’ Law

The Problem

Let us concern ourselves with Disease X. In the United States, the disease prevalence is 12.4 per 100,000 people. This is 0.0124% of the population. There is a new test for this disease, called the Blackbird Test. To determine how good this new test is, a clinical trial was held using 60 subjects with the disease and 40 subjects without the disease. All 100 subjects were tested for the disease using the Blackbird Test. Some tested positive; some tested negative. The following table (confusion matrix) summarizes these clinical results:

Confusion matrix corresponding to the problem given above.
    Reality
    Diseased, D Clean, C
Test Result Positive, + 50 6
  Negative, − 10 34
       

With this information, calculate the probability that a randomly-selected person from the population, who tests positive, has Disease X; that is, calculate P[D | +].

Supplemental Calculations

The following are additional probabilities that are frequently calculated on such clinical data. See if you can calculate them successfully.

Supplemental Calculations
False Positive RateP[+ | C]
   
False Negative RateP[− | D]
   
True Positive RateP[+ | D]
   
True Negative RateP[− | C]
   
SensitivityP[+ | D]
   
SpecificityP[− | C]

You may want to calculate these values yourself then hover your mouse over the grey spaces to see if you calculated them correctly.

Your Answer

In the box below, please enter the probability the person has Disease X, given they test positive for it (P[D | +]), then click on the “Check your answer!” button. Please round your answer to the ten-thousandths place.

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