100 Quant Trading Interview Questions (with Answers)

This is the complete reference list for quant trading interviews. 100 questions across 8 categories, each with a complete answer. Bookmark it, work through it category by category, and use it as a final review before your interview.

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Section 1: Mental Arithmetic (Questions 1–15)

These test pure calculation speed — especially relevant for Optiver (80-in-8) and Jane Street.

1. What is 17 × 13? 221. Use (15+2)(15−2) = 225 − 4 = 221. Or: 17×10 + 17×3 = 170 + 51 = 221.

2. What is 24 × 25? 600. Trick: 25 = 100/4, so 24 × 25 = 24 × 100 / 4 = 2400/4 = 600.

3. What is 18²? 324. Use (20−2)² = 400 − 80 + 4 = 324.

4. What is 125²? 15,625. Use 125 = 1000/8, so 125² = 1,000,000/64 = 15,625.

5. What is 7/8 as a percentage? 87.5%. Know these: 1/8 = 12.5%, so 7/8 = 7 × 12.5% = 87.5%.

6. What is 3/7 as a decimal (to 2 decimal places)? ≈ 0.43. Know: 1/7 ≈ 0.1429, so 3/7 ≈ 0.4286.

7. What is 37 × 43? 1,591. Use (40−3)(40+3) = 1,600 − 9 = 1,591.

8. What is 15% of 240? 36. 10% = 24, 5% = 12. Total: 36.

9. Estimate √150. ≈ 12.25. √144 = 12, √169 = 13. Linear interpolation: 12 + (150−144)/(169−144) = 12 + 6/25 ≈ 12.24.

10. What is 2¹⁰? 1,024. Memorize: 2¹⁰ = 1,024 ≈ 10³.

11. What is 12 × 13 × 14? 2,184. 12 × 13 = 156. 156 × 14 = 156 × 10 + 156 × 4 = 1,560 + 624 = 2,184.

12. What is 63 × 67? 4,221. Use (65−2)(65+2) = 65² − 4 = 4,225 − 4 = 4,221.

13. How many seconds in a year? ≈ 31.5 million. 365 × 24 × 3,600 = 365 × 86,400 ≈ 31,536,000.

14. What is 1/6 + 1/8? 7/24. LCM(6,8) = 24. 4/24 + 3/24 = 7/24.

15. Approximate 1/0.034. ≈ 29.4. 1/0.034 = 1000/34 ≈ 29.4.

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Section 2: Probability Basics (Questions 16–30)

16. A fair coin is flipped 3 times. P(all heads)? (1/2)³ = 1/8 = 12.5%

17. Two dice are rolled. P(sum = 7)? 6/36 = 1/6 ≈ 16.7%. Ways: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1).

18. A card is drawn from a standard deck. P(ace or red card)? P(ace) + P(red) − P(red ace) = 4/52 + 26/52 − 2/52 = 28/52 = 7/13 ≈ 53.8%

19. You flip a coin until you get heads. P(you need exactly 4 flips)? P(TTT then H) = (1/2)³ × (1/2) = (1/2)⁴ = 6.25%

20. You roll a die twice. P(second roll > first roll)? Count: of 36 outcomes, 15 have second > first. P = 15/36 = 5/12 ≈ 41.7%

21. P(at least one 6 in 4 rolls of a fair die)? 1 − P(no 6 in 4 rolls) = 1 − (5/6)⁴ = 1 − 625/1296 ≈ 51.8%

22. A bag has 5 red, 3 blue balls. Draw 2 without replacement. P(both red)? (5/8) × (4/7) = 20/56 = 5/14 ≈ 35.7%

23. P(getting exactly 2 heads in 5 fair coin flips)? C(5,2) × (1/2)² × (1/2)³ = 10 × 1/32 = 10/32 = 31.25%

24. You roll a fair die. Given the result is odd, P(it's > 3)? Odd results: . Those > 3: . P = 1/3

25. Two cards are drawn without replacement. P(both aces)? (4/52) × (3/51) = 12/2652 = 1/221 ≈ 0.45%

26. A fair coin is flipped 10 times. P(exactly 5 heads)? C(10,5)/2¹⁰ = 252/1024 ≈ 24.6%

27. P(sum ≥ 10 when rolling two dice)? Outcomes summing to 10: (4,6),(5,5),(6,4) = 3. To 11: (5,6),(6,5) = 2. To 12: (6,6) = 1. Total = 6. P = 6/36 = 1/6

28. A test is 90% sensitive, 95% specific. Disease prevalence = 2%. You test positive. P(you have disease)? P(+) = 0.9×0.02 + 0.05×0.98 = 0.018 + 0.049 = 0.067. P(disease|+) = 0.018/0.067 ≈ 26.9%

29. P(birthday paradox: at least 2 of 23 people share a birthday)? ≈ 50.7%. The calculation: 1 − (365!/342!)/365²³.

30. You roll a die. P(result is prime)? Primes on a die: . P = 3/6 = 1/2

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Section 3: Expected Value (Questions 31–45)

31. EV of a roll of a fair die? (1+2+3+4+5+6)/6 = 3.5

32. You flip a coin: win $3 heads, lose $1 tails. EV per flip? 0.5×3 + 0.5×(−1) = +$1.00

33. You roll a die and win the face value. What's the Kelly fraction if you can bet on "will be ≥ 4"? P(≥4) = 3/6 = 0.5. EV of "≥4" bet (pays 1:1) = 0.5×1 + 0.5×(−1) = 0. Don't bet (EV = 0).

34. A game costs $5 to play. Roll two dice: win $10 if sum is 7, nothing otherwise. Play? P(sum=7) = 1/6. EV of prize = $10/6 ≈ $1.67. Cost = $5. Don't play: EV = $1.67 − $5 = −$3.33

35. Expected number of flips of a fair coin to get heads? Geometric with p = 0.5. E = 1/0.5 = 2 flips

36. Expected number of rolls of a fair die to get a 6? Geometric with p = 1/6. E = 6 rolls

37. You roll a die and can reroll once (must keep second roll). EV of the game? Reroll iff X ≤ 3. E = (1/2)×3.5 + (1/2)×(4+5+6)/3 = 1.75 + 2.5 = $4.25

38. Expected sum of two fair dice? E[die1] + E[die2] = 3.5 + 3.5 = 7

39. A lottery ticket costs $2. It pays $100 with P = 0.01, $10 with P = 0.05, nothing otherwise. EV? EV = 100×0.01 + 10×0.05 − 2 = 1 + 0.5 − 2 = −$0.50 (don't buy)

40. Expected number of coin flips to get two consecutive heads? Using recursion (see probability-interview-questions guide): 6 flips

41. You're offered a bet: flip a biased coin (P(H) = 0.6). Win $1 on heads, lose $1 on tails. What's the Kelly fraction? f* = 2p − 1 = 2×0.6 − 1 = 20% of bankroll

42. A stock doubles with P = 0.55, halves with P = 0.45. EV after 1 period? E[S₁] = 0.55×2S + 0.45×0.5S = 1.1S + 0.225S = 1.325S (+32.5%)

43. What's the EV of the maximum of two independent uniform [0,1] random variables? E[max(U₁,U₂)] = 2/3. For max of n uniform [0,1]: E = n/(n+1). For n=2: 2/3

44. Expected number of cards to draw from a shuffled deck before seeing an ace? (52+1)/(4+1) = 53/5 = 10.6 cards

45. A market maker quotes a $0.10 bid-ask spread. They trade 50,000 shares/day with 25% informed flow. If informed traders cause $0.15 adverse moves, what's the daily P&L? Revenue = 0.05 × 50,000 = $2,500 (half-spread per trade). Loss to informed = 0.25 × 50,000 × 0.15 = $1,875. Net P&L = $625/day

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Section 4: Conditional Probability and Bayes (Questions 46–55)

46. P(both children are boys | at least one is a boy)? Sample space for 2 children: BB, BG, GB, GG. At least one boy: BB, BG, GB. Both boys: BB. P = 1/3

47. Monty Hall: you pick door 1, host opens door 3 (goat). Should you switch? Yes. P(win by switching) = 2/3. The host's information updates the probability.

48. A bag has 2 gold/gold coins and 1 gold/silver. You draw a coin and see gold. P(other side is gold)? Two gold faces come from G/G (probability 2/3 of all gold faces shown). P = 2/3

49. Drug test: 99% sensitive, 99% specific. 0.5% of applicants use drugs. Someone tests positive. P(they use drugs)? P(+) = 0.99×0.005 + 0.01×0.995 = 0.00495 + 0.00995 = 0.01490. P(drug|+) = 0.00495/0.01490 ≈ 33.2%

50. You have 3 coins: fair, 2-headed, 2-tailed. Pick randomly, flip once — heads. P(2-headed coin)? P(H|2H) = 1, P(H|fair) = 1/2, P(H|2T) = 0. P(H) = 1/3×1 + 1/3×1/2 + 1/3×0 = 1/2. P(2H|H) = (1×1/3)/(1/2) = 2/3

51. P(roll a 6 on die 1 OR roll a 6 on die 2) for two independent dice? P(A∪B) = P(A) + P(B) − P(A∩B) = 1/6 + 1/6 − 1/36 = 12/36 − 1/36 = 11/36 ≈ 30.6%

52. Two events A and B: P(A) = 0.4, P(B) = 0.3, P(A∩B) = 0.1. Are they independent? If independent: P(A∩B) should equal P(A)×P(B) = 0.4×0.3 = 0.12. But P(A∩B) = 0.1 ≠ 0.12. Not independent (negatively correlated).

53. Three people each roll a die. P(all three roll different numbers)? First person: any number (P=1). Second: 5 out of 6 are different (P=5/6). Third: 4 out of 6 (P=4/6). P = 1 × 5/6 × 4/6 = 20/36 = 5/9 ≈ 55.6%

54. An urn has 5 red and 5 blue balls. Draw 3 without replacement. P(exactly 2 red)? C(5,2)×C(5,1)/C(10,3) = 10×5/120 = 50/120 = 5/12 ≈ 41.7%

55. A coin lands on heads 7 of 10 flips. P(it's biased toward heads) if prior is 50/50 between fair and P(H)=0.7? P(7H|fair) = C(10,7)×0.5¹⁰ = 120/1024 ≈ 0.117. P(7H|biased) = C(10,7)×0.7⁷×0.3³ = 120×0.0824×0.027 ≈ 0.267. P(biased|7H) = (0.267×0.5)/(0.267×0.5 + 0.117×0.5) = 0.267/0.384 ≈ 69.5%

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Section 5: Combinatorics and Counting (Questions 56–65)

56. How many ways can 5 people sit in a row? 5! = 120

57. How many ways can you choose 3 people from 10? C(10,3) = 10!/(3!×7!) = 120

58. How many 4-letter passwords use letters A-Z, no repeats? 26 × 25 × 24 × 23 = 358,800

59. P(getting a full house in 5-card poker)? 3,744 / 2,598,960 ≈ 0.144%

60. In how many ways can 4 items be arranged in a circle? (n−1)! = 3! = 6 (circular permutations fix one element)

61. How many distinct 3-digit numbers can be formed from without repetition? 5×4×3 = 60

62. P(at least one pair in a 5-card hand)? Easier to compute the complement: P(no pair) = (52×48×44×40×36)/(52×51×50×49×48) ≈ 50.1%. P(at least one pair) ≈ 49.9% (but including full houses, trips, etc. this is actually closer to 100% − P(all different) = 100% − P(high card) ≈ 50%)

63. A committee of 4 is chosen from 6 men and 4 women. P(at least 2 women)? Total: C(10,4) = 210. P(≥2W): C(4,2)×C(6,2) + C(4,3)×C(6,1) + C(4,4)×C(6,0) = 90+24+1 = 115. P = 115/210 ≈ 54.8%

64. How many ways can 8 people be split into two groups of 4? C(8,4)/2 = 70/2 = 35 (divide by 2 since groups are unordered)

65. A license plate has 3 letters followed by 3 digits. How many possible plates? 26³ × 10³ = 17,576 × 1,000 = 17,576,000

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Section 6: Markets and Finance (Questions 66–80)

66. What is the P/E ratio and what does a P/E of 25 imply? Price-to-Earnings ratio: market cap divided by annual earnings. P/E = 25 implies investors pay $25 for every $1 of earnings — roughly a 4% earnings yield. With required return of 10%, it implies ~6% annual earnings growth in perpetuity.

67. Apple's market cap is approximately $3 trillion. What is its approximate annual revenue? ≈ $380 billion (as of 2024). P/S ratio ≈ 8×.

68. If the Fed raises rates by 25 bps, approximately how much does the value of a 10-year bond change? A 10-year bond has a duration of roughly 9 years. For a 0.25% rate increase: ΔP ≈ −Duration × Δr = −9 × 0.0025 = −2.25% price decline

69. What is the VIX and what level is considered "elevated"? VIX = CBOE Volatility Index, measuring implied 30-day volatility on the S&P 500. "Normal" = 15–20. "Elevated" = 25–30. "Fear" = 30+. During the 2020 COVID crash, VIX hit 80.

70. If a stock has a beta of 1.5, what happens to it when the S&P 500 falls 2%? Expected move: 1.5 × (−2%) = −3% (higher beta = amplified market moves)

71. What is a basis point? 1 bps = 0.01%. 100 bps = 1%. If rates rise from 4.25% to 4.50%, that's a 25 bps increase.

72. What is the rough annual cost of credit card debt at 22% APR on a $10,000 balance? $10,000 × 0.22 = $2,200/year (about $183/month)

73. A $100 stock pays a $2 annual dividend. What is the dividend yield? $2/$100 = 2%

74. If inflation runs at 3% annually, how many years to halve the purchasing power of $1? Rule of 70: 70/3 ≈ 23 years

75. What is the current approximate level of US GDP? ≈ $27–28 trillion (2024). Roughly $80,000 per American.

76. What does it mean for a stock to be "ex-dividend"? When a stock goes ex-dividend, new buyers don't receive the next dividend payment. The stock price typically drops by approximately the dividend amount on the ex-date.

77. What is the approximate market cap of the S&P 500? ≈ $40–45 trillion total. Top 10 stocks make up about 35% of the index.

78. If the 10-year Treasury yield is 4.2%, what's the approximate annual interest cost on the US national debt of $34 trillion? $34T × 0.042 ≈ $1.4 trillion/year in interest payments.

79. What is the difference between a futures contract and a forward contract? Futures: standardized, exchange-traded, daily mark-to-market, cash-settled. Forwards: customized, over-the-counter, settled at expiration, counterparty risk. Both obligate the holder to buy/sell at the agreed price.

80. If a stock option has a delta of 0.4 and the stock moves up $2, approximately how much does the option price change? ΔC ≈ delta × ΔS = 0.4 × $2 = +$0.80

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Section 7: Brainteasers and Logic (Questions 81–92)

81. You have 9 balls, identical except one is slightly heavier. How many weighings on a balance scale to find it? 2 weighings. Split into three groups of 3. Weigh group 1 vs group 2. If balanced: heavy ball is in group 3 (weigh two of the three). If unbalanced: heavy ball is in the heavier group (weigh two of the three in that group).

82. A clock shows 3:15. What is the angle between the hour and minute hands? Minute hand at 3:15 = 90°. Hour hand at 3:15 = 90° + (15/60)×30° = 90° + 7.5° = 97.5°. Angle = 7.5°

83. You have a 3-gallon jug and a 5-gallon jug. How do you measure exactly 4 gallons? Fill 5-gallon. Pour into 3-gallon (fills it). 2 gallons remain in 5-gallon. Empty 3-gallon. Pour 2 gallons into 3-gallon. Fill 5-gallon again. Pour 5-gallon into 3-gallon (which already has 2 gallons, so takes 1 more). 4 gallons remain in the 5-gallon jug.

84. Three ants sit at corners of an equilateral triangle. Each randomly picks a direction and walks along an edge. P(no collision)? Each ant independently chooses clockwise or counterclockwise (P=1/2 each). No collision iff all go clockwise or all go counterclockwise. P = (1/2)³ + (1/2)³ = 1/4 = 25%

85. You have 12 balls, one different weight (heavier or lighter). Minimum weighings to find it and determine if it's heavier or lighter? 3 weighings. 3³ = 27 outcomes covers 24 possibilities (12 balls × heavier/lighter). The strategy exists and is provably optimal.

86. P(a random chord of a circle is longer than the radius)? This is Bertrand's paradox — the answer depends on how you define "random." Under the "random endpoints" method: P = 1/3. Under "random midpoint": P = 1/4. The question has no single answer without specifying the distribution.

87. You enter a room with 3 light switches and 3 light bulbs in another room. You can only enter the bulb room once. How do you identify which switch controls which bulb? Turn on switch 1 for 5 minutes, then off. Turn on switch 2. Enter the room. Switch 2's bulb is on. Switch 1's bulb is off but warm. Switch 3's bulb is off and cold.

88. Two people are born in the same year. P(same birthday)? Assuming uniform distribution: 1/365 ≈ 0.27%. (Not the birthday paradox, which asks about a group.)

89. A train leaves City A at 60 mph. Another leaves City B at 40 mph, 200 miles away, heading toward A. When do they meet? Closing speed = 100 mph. Time = 200/100 = 2 hours. Distance from A: 60 × 2 = 120 miles.

90. You have a fair coin. How do you simulate a fair die using coin flips? Flip 3 coins for a binary encoding of 0–7. If result is 0–5, map to 1–6 (mod 6 + 1). If result is 6 or 7 (probability 2/8), flip again. Expected flips ≈ 3/(6/8) = 4 flips.

91. 100 people board a plane. Person 1 has lost their boarding pass and sits randomly. Each subsequent person sits in their assigned seat if available, or randomly otherwise. P(person 100 sits in their own seat)? 1/2 = 50%. Classic result: the last passenger either ends up in seat 1 or seat 100, with equal probability.

92. What is the sum of integers from 1 to 100? Gauss's formula: n(n+1)/2 = 100×101/2 = 5,050

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Section 8: Options and Derivatives (Questions 93–100)

93. What is a call option? What's the payoff at expiration? A call option gives the right (not obligation) to buy the underlying at the strike price K. Payoff at expiration: max(S_T − K, 0) where S_T is the stock price at expiration.

94. If a call has delta = 0.6 and gamma = 0.05, and the stock rises $3, what's the approximate new delta? New delta ≈ 0.6 + 0.05 × 3 = 0.75

95. What does implied volatility (IV) represent? IV is the market's expectation of future volatility, backed out from option prices using Black-Scholes. If IV = 30%, the market expects the stock to move about 30%/√252 ≈ 1.9% per day.

96. Why do out-of-the-money puts on equity indexes tend to have higher IV than at-the-money puts? (Vol smile/skew) The "vol skew" exists because: (1) investors buy OTM puts as insurance against crashes, bidding up their price/IV; (2) markets historically exhibit negative skewness (large down moves more common than large up moves); (3) leverage effect (falling stock prices increase volatility).

97. What is put-call parity? C − P = S − PV(K), where C = call price, P = put price, S = stock price, PV(K) = present value of strike. Equivalently: C − P = S − Ke^(-rT). Arbitrage enforces this relationship.

98. A stock is at $100. A 1-year call with strike $100 costs $10. Risk-free rate = 5%. What's the put price? Put-call parity: P = C − S + Ke^(-rT) = 10 − 100 + 100×e^(-0.05) = 10 − 100 + 95.12 = $5.12

99. What is vega and when does it matter most? Vega = sensitivity of option price to a 1% change in implied volatility. Options with high vega: long-dated, at-the-money options. Vega matters most when you're trading volatility rather than direction — e.g., straddles, earnings trades, or volatility ETF positions.

100. If you're long a straddle (long call + long put at the same strike), when do you profit? You profit when the stock moves significantly in either direction — more than the premium paid for both options. You lose if the stock stays near the strike (both options expire near worthless). It's a bet on realized volatility > implied volatility.

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Using This List

Work through these questions in order, timed. Target:

• Mental arithmetic (Section 1): under 20 seconds per question
• Probability and EV (Sections 2–4): under 2 minutes per question
• Markets and options (Sections 6, 8): under 1 minute per question

Then shuffle them. The hardest skill isn't answering any single question — it's rapidly categorizing what type of problem you're facing and applying the right technique.

Practice the underlying skills daily at Fermiq. The estimation and probability drills build the fluency that makes these questions automatic rather than effortful.