Trading OA — 15-Question Probability Quickfire (30 Minutes)

Citadel Trading OA format: 15 probability / expected-value questions in 30 minutes, averaging 2 minutes per question. Standard quant-interview probability staples (color-pair runs, uniform-pair differences, binary-choice population balancing) at a pace that rewards pattern recognition over derivation. Requirements Non-interactive timed multiple-choice / numeric-input quiz on HackerRank or a similar platform. 15 questions, 30-minute total time limit. No code; pure pen-and-paper probability and combinatorics. Examples Examples are reproduced verbatim from one reported sitting. 6 red cards and 4 black cards are shuffled randomly. What is the expected number of adjacent pairs of cards having the same color? 5 people visit one of two places, chosen independently and uniformly at random. What is the expected number of people at the less-populated place? X and Y are random real numbers independently and uniformly picked between 0 and 100. What is the probability that X and Y differ by 50 to 80? The rest of the 15 questions span the same probability-puzzle family (expected value, conditional probability, simple combinatorics, geometric probability on the unit square / cube). Notes Expected adjacent same-color pairs: use indicator random variables. For each of the 9 adjacent pair positions, compute P(same color); sum the probabilities. For 6 red + 4 black: each pair is same-color with probability (C(6,2) + C(4,2)) / C(10,2) = (15 + 6) / 45 = 21/45 = 7/15. Expected total = 9 * 7/15 = 21/5 = 4.2. Expected count at less-populated of two places with 5 people: by symmetry, partition the count into (0,5), (1,4), (2,3) cases with binomial probabilities 1/32, 5/32, 10/32 each side (1/16, 5/16, 10/16 after combining both sides); the less-populated count is min(k, 5-k). Compute weighted average: 2 * (0 * 1/32 + 1 * 5/32 + 2 * 10/32) = 2 * 25/32 = 50/32 = 25/16 ≈ 1.5625. Uniform pair difference 50 <= |X - Y| <= 80: the region in the [0,100] x [0,100] square between the two bands |X - Y| = 50 and |X - Y| = 80. Compute the area of the region with |X - Y| <= 80 (square minus two corner triangles) minus the area with |X - Y| <= 50. Final probability is (area in band) / 10000. Pace discipline: 2 minutes per question. Skip and return; do not over-invest. Most questions reward setting up the indicator-sum or the symmetry argument quickly, not a full Bayesian derivation. Common slip: trying to enumerate cases instead of using indicator-sum linearity. The expected-value questions almost always collapse via linearity. Preparation Drill the indicator-random-variable pattern until it is automatic. It unlocks half the typical Citadel / quant trading probability question bank. Run through a classic probability-puzzle compendium (50 Challenging Problems in Probability, Heard on the Street, A Practical Guide to Quantitative Finance Interviews) front to back. Quickfire OAs sample heavily from the same canon. Practice symmetry arguments and conditional-probability tree drawing on paper at 90-second pace. The mental shortcut beats deriving from first principles every time. Brush up on geometric probability on [0,1]^2 and [0,1]^3 (uniform pair / triple differences, triangle inequality probabilities, Buffon-style integrations). These appear repeatedly.