This is the bread and butter of trading interviews. You are tested on your ability to price risk and determine the "fair value" of a game.
Classic Questions:
The Insight: Interviewers are looking for the concept of Expected Value (EV) and Game Theory. In the dice game, you should re-roll if the first result is less than the expected value of a single roll (3.5). So, you keep 4, 5, and 6. This changes the calculation for the total value of the game. 150 Most Frequently Asked Questions On Quant Interviews
Core concepts: Conditional probability, Bayes’ theorem, expectation, variance, distributions.
| # | Question | Difficulty | Key Idea | |---|----------|------------|-----------| | 1 | You flip two fair coins. Given that at least one is heads, what is the probability both are heads? | ★ | Conditional probability: 1/3 | | 2 | Draw one card from a deck. What is the probability it is a king or a heart? | ★ | Inclusion-exclusion: 4/52 + 13/52 – 1/52 = 16/52 | | 3 | Roll a fair die. What is the expected value? | ★ | (1+2+3+4+5+6)/6 = 3.5 | | 4 | You have two dice. What is the probability the sum is 7? | ★ | 6/36 = 1/6 | | 5 | A family has two children. At least one is a boy. Probability both are boys? | ★ | 1/3 | | 6 | You flip a coin until you get heads. Expected number of flips? | ★ | Geometric: 2 | | 7 | Draw two cards without replacement. Probability both are aces? | ★ | (4/52)(3/51) | | 8 | You roll a die. What is the variance of the outcome? | ★★ | E[X²] – E[X]² = 91/6 – 3.5² ≈ 2.92 | | 9 | You flip a fair coin 10 times. Probability of exactly 5 heads? | ★ | C(10,5)/2¹⁰ | | 10 | There are 3 red and 3 blue balls in an urn. Draw two without replacement. Probability same color? | ★ | (3/62/5)2 = 2/5 | | 11 | You have two coins: one fair, one double-headed. Pick one at random, flip, get heads. Probability it’s the double-headed? | ★★ | Bayes: 2/3 | | 12 | What is the expected number of rolls of a die to see all 6 faces? | ★★ | Coupon collector: 6(1 + 1/2 + … + 1/6) ≈ 14.7 | | 13 | You and I take turns flipping a coin. First to get heads wins. You go first. Your chance to win? | ★★ | 2/3 | | 14 | Two points are chosen uniformly on [0,1]. Expected distance between them? | ★★ | 1/3 | | 15 | Random variable X ~ N(0,1). What is E[|X|]? | ★★ | √(2/π) | | 16 | You have a stick of length 1. Break at random point. Expected length of shorter piece? | ★★ | 1/4 | | 17 | 100 people randomly assigned seats on a plane. First person sits randomly. Probability last person gets own seat? | ★★★ | 1/2 (symmetry) | | 18 | You flip a fair coin until you see HH. Expected flips? | ★★★ | 6 (use Markov chains) | | 19 | You have n biased coins with p_i. Randomly pick one, flip. Probability heads? | ★ | Average of p_i | | 20 | Roll two dice. Expected maximum? | ★★ | ≈ 4.472 | | 21 | Draw from U[0,1] until sum exceeds 1. Expected number of draws? | ★★★ | e ≈ 2.718 | | 22 | What is the probability that a random chord in a circle is longer than the radius? | ★★ | 1/2 (depends on definition) | | 23 | You have 5 red and 5 blue balls. Draw without replacement. Probability last ball is red? | ★ | 1/2 (symmetry) | | 24 | You roll a die and win $1 if prime, lose $1 if composite, 0 otherwise. Expected profit? | ★ | (3 wins, 2 losses, 1 zero) → 1/6 | | 25 | Random permutation of n numbers. Probability that 1 is before 2? | ★ | 1/2 | | 26 | What is the probability of getting a flush (5 same suit) in poker? | ★★ | (4*C(13,5))/C(52,5) | | 27 | You have two envelopes with money, one double the other. You open one, see $100. Switch? | ★★ | Paradox: Expected value same | | 28 | You play a game: roll a die, get that many dollars. You can roll again once. Strategy? | ★★ | Roll again if ≤ 3 | | 29 | Random walk on integers starting at 0. Probability of reaching +1 before –n? | ★★ | 1/(n+1) | | 30 | You have 3 doors, one car. Pick one, host opens a goat door. Switch? | ★ | Switch gives 2/3 chance | | 31 | Random point in a unit square. Expected distance to nearest edge? | ★★ | 1/6 | | 32 | What is P(X < Y) for independent exponentials with rates λ, μ? | ★ | λ/(λ+μ) | | 33 | You flip a coin until you get HT. Expected flips? | ★★★ | 4 | | 34 | Two players shoot basketball with accuracy p. Alternate. First to make wins. Advantage to first? | ★★ | 1/(2-p) | | 35 | Roll a die. If you get 6 you win $6, else roll again. Expected value? | ★★ | 3.5 if stop otherwise? | This is the bread and butter of trading interviews
For front-office quant roles, you must know the "Greeks" and the Black-Scholes model. This is where the heavy mathematics comes into play.
Frequent Topics:
The Insight: You are expected to understand the relationship between volatility, time decay (Theta), and the underlying asset price. A common trick question involves intuitive pricing: "If volatility doubles, does the price of the call option double?" (Answer: No, it increases by roughly $\sqrt2$ due to the square root of time rule in volatility scaling).
