Quant OA Math / Probability Multiple Choice Bank

Goldman quant / engineering OAs include multiple-choice calculus, probability, and statistics questions alongside coding. Recent examples cover unit-circle optimization, Laplace-style density probabilities, conditional card probabilities, dice expectations, normal standard errors, and coin-run expectations. Requirements Practice these prompt shapes as short multiple-choice / numeric-answer drills: Unit circle optimization: P = (Px, Py) is the first-quadrant point on x² + y² = 1 maximizing f(x,y)=4x²y; find Py². Continuous density: f(x)=0.5e^{-|x|} over all real x; find P(2 <= |X| <= 4) rounded to three decimals. Conditional cards: a 5-card hand is known to contain at least 3 aces; find the probability it contains all 4 aces. Dice sum: roll a fair die 5 times; find the probability the sum is a multiple of 5. Discrete distribution normalization: for R_X={1,2,...,12} and f(x)=k(2x-1), find k. Coin runs: toss a fair coin 9 times; find the expected number of consecutive same-outcome runs. Standard error: X1...X100 are iid standard normal, Y is their average; find the standard deviation of Y. Three-coin game: two players each flip 3 coins; if they have the same number of heads, payout is 2^n, where n is the number of heads. Find expected value. Dice max: roll three ordinary dice and score the largest face; find the probability the score is 2. Notes These are not deep quant puzzles; the speed bar is recognizing standard distributions, conditioning, and expectation linearity under time pressure. The OA can mix these math questions with two coding problems in a 180-minute window. Preparation Drill conditional probability with counting formulas, especially poker-hand conditions and dice / coin event partitions. Re-derive standard error of the mean and expectation of runs without notes. For calculus optimization, practice Lagrange multipliers and one-variable substitution on the unit circle.