Doctors have devised a test for leptospirosis that has the following property:…

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Doctors have devised a test for leptospirosis that has the following property: For any person suffering from lepto, there is a 90% chance of the test returning positive. For a person not suffering from lepto, there is an 80% chance of the test returning negative. It is known that 10% of people who go for testing have lepto. If a person who gets tested gets a +ve result for lepto (as in, the test result says they have got lepto), what is the probability that they actually have lepto?

  1. A.

    7/10

  2. B.

    8/11

  3. C.

    1/3

  4. D.

    1/2

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Correct answer: C

Concept: Bayes' theorem converts a test's known accuracy (sensitivity and specificity) into the actual post-test probability of disease, by combining it with the disease's prior probability (prevalence): P(Disease | Positive) = [P(Positive | Disease) x P(Disease)] / P(Positive), where the denominator P(Positive) is found via the law of total probability, summing the positive-test probability over both the diseased and healthy groups. This weighting by prevalence is essential: a fairly accurate test can still have a low post-test probability of disease when the disease itself is rare among those tested.

Application: Applying this to the given numbers:

  • P(has lepto) = 0.1 (10% prevalence)

  • P(no lepto) = 0.9

  • P(positive | has lepto) = 0.9 (sensitivity)

  • P(positive | no lepto) = 1 minus 0.8 = 0.2 (false-positive rate, from 80% specificity)

  1. Find the overall probability of a positive test using the law of total probability: P(+) = P(has lepto)xP(+|has lepto) + P(no lepto)xP(+|no lepto) = 0.1x0.9 + 0.9x0.2 = 0.09 + 0.18 = 0.27.

  2. Apply Bayes' theorem: P(has lepto | +) = P(+|has lepto)xP(has lepto) / P(+) = 0.09 / 0.27 = 1/3 approx 0.333.

Cross-check: Confirm with a natural-frequency check on 1000 tested people: 100 actually have lepto (10%), of whom 90 test positive (90% sensitivity); of the 900 without lepto, 20% test positive, i.e. 180 false positives. Total positives = 90 + 180 = 270, of which only 90 truly have lepto, 90/270 = 1/3, matching the Bayes' theorem result.

Interpretation: Even with a fairly accurate test, a positive result gives only a one-in-three chance of actual disease here, because the disease is relatively rare among those tested.

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