Consider a rare disease that affects 1 in every 10,000 people. A diagnostic test
for this disease has the following characteristics:
• The probability of the test returning a positive result if the person actually
has the disease (True Positive rate) is 99%.
• The probability of the test returning a positive result if the person does
not have the disease (False Positive rate) is 1%.
Question:
(a) If a person tests positive, what is the probability that they actually have
the disease? Show your work using Bayes’ Rule.
(b) Discuss why the probability calculated in part (a) might be surprising to
someone who is not familiar with Bayes’ Rule. What does this tell us
about the reliability of medical tests for rare diseases?
Hints:
Let D represent the event that the person has the disease. Let + represent the
event that the test is positive. Use Bayes’ Rule, which states:
P (D | +) = P (+ | D) · P (D)
P (+)
To find P (+), use the law of total probability:
P (+) = P (+ | D) · P (D) + P (+ | Dc) · P (Dc)
where Dc is the complement of D, representing the event that the person does
not have the disease.
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