Beat the Odds — Probability Test

The signature Optiver quant assessment: 30 multiple-choice probability questions, ~90 seconds each, no going back. You pick the answer closest to the true value, so speed and estimation matter more than exact computation. Wrong answers are penalized. Requirements 30 questions (QT/QR rotation; some sittings report 10–20), roughly 90 seconds each. You cannot return to a skipped question, but you may skip. Each question is multiple choice (usually 5 options) and you select the value closest to the true answer — exact arithmetic is often unnecessary, fast estimation wins. Wrong answers carry a penalty, so blind guessing is not free. Topics: dice/coin/card probability, expected value, symmetric random walks on polygons (hitting / return time), gambler's ruin, combinatorics, central-limit-style estimation, and optimal-stopping flavored questions. The pool is a rotating question bank; identical question types (and sometimes the same question) recur within one sitting and across sittings, so drilling the bank pays off directly. Programming-minded candidates often pre-simulate the recurring families. Examples Verbatim questions candidates have reported: You throw one die twice. What is the probability the 2nd throw has a different face value than the first? (closest to 0.75) Take a shuffled 52-card deck and throw away the top 10 cards. What is the probability the new top card is red? (0.5) You throw two dice. Probability they add up to 11 or 12? (~0.1) Flip a coin 3 times — probability the outcome is the same for all flips? (0.25) 61 coins randomly placed into 15 boxes; you win if any box has more than 4. Probability you win? (1) A bag has 10 red, 10 yellow, 10 blue balls. Draw until only two colours remain. Expected number of balls remaining? Roll a die, keeping a running sum, until the sum exceeds 100. Probability the last roll was a 2? Two particles at opposite vertices of an octagon; each second each flips a coin to move clockwise/anticlockwise one edge. Expected number of coin flips until they meet? A particle on a vertex of a decagon: each second roll a die — 1/2 move clockwise, 3/4 anticlockwise, 5/6 stay. Expected steps to reach the initially-farthest vertex. Particle at the centre of a 10×10 grid moving up/right/down/left with probability 1/4 each — expected steps to hit the boundary. Start with $10, bet $1 per round on a fair (or 60/40) coin, stop at $20 or $0 — probability you reach $20 (gambler's ruin). Flip 6 coins — probability of at most 3 heads / more heads than tails. A machine outputs integers uniformly 1–10. You and a friend each take the sum of two draws — probability the sums are equal. Integer chosen uniformly from 1 to 1,000,000 — probability it contains the digit 7 at least once. Notes The recurring structural families are worth memorizing as closed forms: symmetric random walk return time on an n-gon (expected return time = n), polygon hitting times via Markov-chain / first-step analysis, gambler's ruin with biased coins, and "remove matched pairs/sets until one type remains" expectations (symmetry + CLT estimate). Several candidates note the 2025/26 pool got noticeably harder than earlier years and that there often is not enough time to compute exactly — the closest-answer format is a deliberate test of numerical intuition. Have a calculator and scratch paper ready; many report typing furiously on a calculator. Some sittings allow pen-and-paper, others are pure on-screen. Preparation Drill the standard green-book and tradinginterview.com probability sets until each family is automatic; build a one-page cheat sheet of closed forms (random-walk return/hitting times, gambler's ruin, coupon-collector, linearity-of-expectation tricks). Write quick simulators for the recurring families (polygon walks, dice running-sum, pair-removal) so you can pre-compute answers and recognize the type in seconds. Practice estimating to the nearest option under a 90-second clock rather than solving exactly — train the "which bucket" reflex.