How does Bayes theorem work? Explain with a medical test example.
Bayes theorem updates a probability using new evidence: P(A|B) = P(B|A) x P(A) / P(B). It shows that a positive result from an accurate test can still mean a low chance of disease when the disease is rare.
Explain it simply
A test says "positive". What you want is the chance you are actually ill, but what the lab quotes is the chance of a positive when you are ill. These are different, and the gap depends on how common the disease is (the prior).
Easiest way to compute: use counts, not formulas.
Example
A disease affects 1% of people. A test catches 90% of sick people (sensitivity) and wrongly flags 5% of healthy people (false positive rate). Take 10,000 people:
- Sick: 100, of whom 90 test positive
- Healthy: 9,900, of whom 5% = 495 test positive
- All positives: 90 + 495 = 585
P(sick | positive) = 90 / 585 = 0.154, about 15%.
sick, healthy = 100, 9900
tp = sick * 0.90
fp = healthy * 0.05
print(round(tp / (tp + fp), 3)) # 0.154Even with a good test, about 85% of positives are false alarms, because healthy people vastly outnumber sick ones.
What to say in the interview
"Bayes theorem reverses a conditional probability: P(A given B) equals P(B given A) times the prior P(A), divided by P(B). The base rate matters. I usually work with natural frequencies, like 10,000 people, to avoid the base-rate fallacy. The same idea underlies spam filters and Naive Bayes classifiers."
Common mistakes
- Treating P(positive | sick) as if it were P(sick | positive).