Data Science interview question · Probability · mid level
A test is 99% accurate and you test positive. What is the chance you have the disease?
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Why it is asked
This is the classic base-rate question; it tests whether you ask for the prevalence instead of answering 99%.
The answer
It depends on how common the disease is; if 1% of people have it and the test is 99% sensitive and 99% specific, the chance is only 50%.
First I would ask what '99% accurate' means. Assume sensitivity = 99% (it catches 99% of sick people) and specificity = 99% (it correctly clears 99% of healthy people), and a prevalence (base rate) of 1%.
Work it out for 10,000 people:
- 100 are sick, and the test catches 99 of them (true positives).
- 9,900 are healthy, and 1% of them, 99 people, test positive anyway (false positives).
- So 99 + 99 = 198 people test positive, and only 99 of them are sick: 99 / 198 = 50%.
The answer moves a lot with the base rate. With the same test:
- prevalence 0.1%: P(sick | positive), the chance of being sick given a positive test, = 9.0%;
- prevalence 10%: P(sick | positive) = 91.7%.
The lesson for data science: precision (the share of flagged cases that are real) depends on the base rate. A fraud model with 99% specificity still produces mostly false alarms if only 0.1% of transactions are fraud. That is why you report precision at the real base rate, not just accuracy or ROC-AUC (a ranking-quality score).
What the interviewer listens for
- Ask for the base rate and what 'accurate' means.
- Count true and false positives in a concrete population.
- Prevalence 1%: 99 true vs 99 false positives, so 50%.
- Precision collapses when the positive class is rare.
Follow-ups the interviewer may ask
- How would a second independent positive test change it? Hint: prior 50% becomes 99 x 0.99 / (99 x 0.99 + 99 x 0.01) = 99%.
- What metric would you show a fraud team? Hint: precision and recall at the operating threshold, at the real fraud rate.
- Why is accuracy useless here? Hint: a test that says 'healthy' to everyone is 99% accurate at 1% prevalence.
Common mistakes
- Answering 99%.
- Not asking for prevalence.
- Treating sensitivity and P(disease | positive) as the same thing.
Written by Roni Das, who builds distributed systems and AI systems for a living.
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