Maths for ML interview question · Probability: Bayes' theorem · mid level
A disease affects 1% of people. A test catches 90% of sick people (sensitivity) and wrongly flags 9% of healthy people (false positive rate). You test positive. What is the chance you are sick?
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Why it is asked
It is the most asked Bayes question. The interviewer checks whether you anchor on the 90% or work through the 1% prior.
The answer
About 9.2%, not 90%. Most positives come from the huge healthy group, so a positive result is still more likely a false alarm than a real case.
Terms: sensitivity = P(test positive | sick). False positive rate = P(test positive | healthy). Specificity = P(test negative | healthy) = 1 - false positive rate = 0.91. Prevalence = P(sick) before any test, which is the prior.
The clearest way is natural frequencies. Imagine 10,000 people:
- Sick: 1% of 10,000 = 100 people. The test catches 90% of them: 90 true positives (10 are missed).
- Healthy: 9,900 people. 9% of them test positive anyway: 891 false positives.
- Total positives: 90 + 891 = 981.
- P(sick | positive) = 90 / 981 ≈ 0.0917, about 9.2%.
The same with Bayes' theorem: P(sick | +) = P(+ | sick) × P(sick) / P(+) = (0.90 × 0.01) / (0.90 × 0.01 + 0.09 × 0.99) = 0.009 / 0.0981 ≈ 0.0917.
The test still did its job. Belief rose from 1% to 9.2%, about 9 times higher. It is a screening result that calls for a better second test, not a diagnosis.
A negative result is very reassuring: P(sick | negative) = 10 / (10 + 9,009) = 10 / 9,019 ≈ 0.0011, about 0.11%.
The same pattern appears in fraud models, content moderation and any alert on a rare event: when the condition is rare, even a good test produces mostly false alarms.
What the interviewer listens for
- Use the prior: 1% prevalence.
- 10,000 people: 90 true positives vs 891 false positives.
- P(sick | +) = 90 / 981 ≈ 0.0917.
- A negative is very reassuring: ≈ 0.0011.
Follow-ups the interviewer may ask
- What if prevalence were 10%? Hint: 0.09 / (0.09 + 0.081) ≈ 0.5263.
- Which number should improve to raise the posterior most? Hint: the false positive rate, because the healthy group is 99 times larger.
- What is the specificity here? Hint: 1 - 0.09 = 0.91.
Common mistakes
- Answering 90% (confusing P(+ | sick) with P(sick | +)).
- Answering 91% from the specificity.
Written by Roni Das, who builds distributed systems and AI systems for a living.
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This is 1 of the 120 questions in the Maths for ML and Deep Learning pack
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