Bayes questions are common because they reliably catch people who compute correctly but set up wrongly.
The statement
P(H | E) = P(E | H) P(H) / P(E)
with P(E) usually expanded by conditioning:
P(E) = P(E | H) P(H) + P(E | not H) P(not H)
Do not use percentages
The single best practical tip: convert to natural frequencies and imagine a concrete population. Percentages invite errors; counts do not.
The canonical problem
A disease affects 1 in 1,000. A test is 99% accurate in both directions. You test positive. What is the probability you have it?
Take 100,000 people.
- 100 have the disease. 99 of them test positive.
- 99,900 do not. 1% of them - 999 people - test positive anyway.
Positives total 99 + 999 = 1,098, of whom 99 actually have it.
P(disease | positive) = 99 / 1098 ≈ 9%
Most people say 99%. The answer is under 10%, because the false positives from the enormous healthy group swamp the true positives from the tiny sick one.
Why it works
When the base rate is small, P(H | E) is driven by the ratio of true positives to false positives, and the false-positive pool is drawn from a much larger population. A test being "99% accurate" tells you P(E | H). It does not tell you P(H | E), and the gap between those is the entire question.
The interview variants
The same structure appears wearing different clothes: a biased coin drawn from a bag of fair ones, a machine that is occasionally faulty, a trader who has been right five times running. The move is always the same - set up a population, count both paths to the evidence, take the ratio.
The one to watch
Interviewers often follow up with "and if you test positive twice?" You cannot simply square anything unless the tests are conditionally independent given disease status, which is worth stating explicitly rather than assuming silently.
Work these in the conditional probability set.