Bayes' Theorem in quant interviews
The machinery for updating beliefs. A staple of quant interviews, often in disguised form.
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In the classical secretary problem, as n → ∞, what fraction should you reject outright before accepting the next best-so-far candidate?
WHAT IT IS
Bayes' theorem states P(H | E) = P(E | H) P(H) / P(E), where H is a hypothesis and E is the evidence observed. In plain words: the probability of the hypothesis given the evidence equals the likelihood of the evidence under the hypothesis, times the prior belief in the hypothesis, divided by the overall probability of the evidence. Bayes is the formal apparatus for all 'now that I've seen X, what should I believe about Y' questions — medical tests, biased coins, population mixtures, spam filters. In interviews it shows up either as a named-Bayes problem or, more commonly, as an unstated-Bayes problem where the candidate has to recognise that Bayes is the right tool. The second form trips people up; the give-away is any question of the form 'given that I observed X, what is the probability of Y?'
WHEN IT APPEARS IN INTERVIEWS
Frequent across all quant firms. Citadel and Jane Street tend to go deeper (with follow-up questions about prior sensitivity or sequential Bayes updates); SIG, Optiver, and IMC use shorter classic Bayes problems (medical test, biased coin jar) as sanity checks on probabilistic reasoning.
FIRMS THAT TEST THIS
SAMPLE PROBLEMS
A jar contains 100 coins: 99 are fair, 1 is two-headed. You pick one at random and flip it 10 times, getting 10 heads. What's the probability you picked the two-headed coin?
Prior: P(2H) = 0.01, P(fair) = 0.99. Likelihood: P(10 heads | 2H) = 1, P(10 heads | fair) = 1/1024. Posterior: (0.01 × 1) / (0.01 × 1 + 0.99 × 1/1024) ≈ 0.912. Counter-intuitively high because the likelihood ratio (1024) dominates the prior (99:1 against).
A disease affects 1 in 10,000 people. A test is 99.9% accurate (both sensitivity and specificity). You test positive. What's the probability you actually have it?
P(disease | +) = (0.001 × 0.0001) / (0.999 × 0.0001 + 0.001 × 0.9999) ≈ 9.1%. The low base rate makes false positives dominate true positives even with a 99.9% test — the core lesson about base-rate neglect.
An urn contains 10 balls, each independently red or blue with probability 1/2. You draw 3 balls without replacement and they're all red. What's the expected number of red balls remaining in the urn?
Update the distribution of the number of red balls via Bayes, then compute the posterior expected count of reds minus 3 (the ones drawn). Uses conjugate-like reasoning — each red draw shifts the posterior toward more-red compositions.
SOLVING STRATEGIES
- ·Start by writing the formula P(H | E) = P(E | H) P(H) / P(E). It's short enough to not skip.
- ·Always compute P(E) as a weighted sum: P(E) = P(E | H) P(H) + P(E | ¬H) P(¬H). Most mistakes happen in the denominator.
- ·Use odds form when numbers are awkward: posterior odds = likelihood ratio × prior odds.
- ·Expect base-rate neglect. If the prior is tiny and the test is imperfect, the posterior is usually smaller than intuition suggests.
- ·For sequential updates, the posterior after evidence E₁ becomes the prior for E₂. This chains cleanly.
COMMON VARIATIONS
- ·Sequential Bayes: updating step-by-step as each piece of evidence arrives.
- ·Multi-hypothesis Bayes (more than two hypotheses) — expand P(E) over all hypotheses.
- ·Continuous Bayes: priors and likelihoods as densities (beta-binomial conjugate, gaussian conjugate).
- ·Odds-ratio form: useful for quick mental updates in trading contexts.
FAQ
Use Bayes when the problem gives you P(E | H) but asks for P(H | E) — in other words, when the direction needs to be flipped. If the conditioning direction is already what you need, plain conditional probability suffices.
Human intuition over-weights the likelihood (how accurate is the test?) and under-weights the prior (how rare is the condition?). Interviewers test this because it's a genuine reasoning failure in real-world settings — trading, medicine, fraud detection — not just an interview gotcha.
For most trading and interview contexts, no. For quant research roles at firms like Citadel or Jane Street, beta-binomial and gaussian conjugacy come up occasionally and are worth a quick review.
Each Bayes problem in the corpus has both a canonical solution and an AI-generated alternative — often the alternative is the odds-ratio framing, which is faster mentally than the formula-based calculation.
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