Why Probability Dominates Quant Interviews
Roughly half of every trader-track quant interview is probability. The questions look elementary - "you flip a coin until..." or "two players take turns rolling..." - but the answers reward fluency with a specific small set of techniques: conditional expectation, recursion on states, Bayes' rule, generating functions, and optional stopping. The candidates who do well are not necessarily the ones with the deepest probability knowledge; they are the ones who recognise problem types in seconds and apply the right technique without hesitation.
This guide collects 30 worked examples drawn from real recent SIG, Jane Street, Optiver, IMC, Citadel and Hudson River Trading interviews. For broader context on probability theory, see our probability for quant finance guide. For more brain-teaser style questions, see our quant brain teasers guide.
Section 1: Expected Value (Questions 1-7)
1. Coin sequence
Flip a fair coin until you get HTH. What is the expected number of flips?
Answer: 10. Set up states by what you have just flipped. Let (E_0) be from the start, (E_H) after seeing H, (E_{HT}) after HT. Then (E_0 = 1 + 0.5 E_H + 0.5 E_0), (E_H = 1 + 0.5 E_H + 0.5 E_{HT}), (E_{HT} = 1 + 0.5 \cdot 0 + 0.5 E_0). Solving: (E_0 = 10).
2. Two consecutive heads
Flip a fair coin until you get HH. What is the expected number of flips?
Answer: 6. (E_0 = 1 + 0.5 E_H + 0.5 E_0), (E_H = 1 + 0.5 \cdot 0 + 0.5 E_0). Solving: (E_0 = 6).
3. Optimal die strategy
Roll a fair six-sided die. You can keep the value or reroll, but only once. What strategy maximises expected value, and what is it?
Answer: 4.25. Reroll if your first roll is below the expected value of a fresh roll (3.5). So reroll on 1, 2, 3 and keep 4, 5, 6. (E = 0.5 \cdot 3.5 + 0.5 \cdot 5 = 4.25).
4. Three rolls
Roll a die three times sequentially; you can stop and take any value, but you cannot go back. What is the expected value of the optimal strategy?
Answer: ~4.67. Backwards induction. On the third roll, take it (expected 3.5). On the second, take it if greater than 3.5; else take the third. (E_2 = 0.5 \cdot 5 + 0.5 \cdot 3.5 = 4.25). On the first, take if greater than 4.25; else take (E_2). (E_1 = (2/6)(5) + (4/6)(4.25) = 4.667).
5. Sum of dice
Roll two dice. What is the expected value of the maximum?
Answer: 161/36 ≈ 4.47. (P(\max = k) = (2k-1)/36) for (k = 1..6). Sum (\sum k(2k-1)/36 = 161/36).
6. Geometric expectation
A coin lands heads with probability p. What is the expected number of flips to see a head?
Answer: 1/p. Standard geometric. For p = 0.3, expected flips = 3.33.
7. Sum to 100
You add fair uniform [0,1] random variables until the sum exceeds 1. What is the expected number of variables added?
Answer: e ≈ 2.718. A classic. The probability that exactly n uniform variables sum to less than 1 is 1/n!, so the expected stopping time is (\sum 1/n! = e).
Section 2: Conditional Probability and Bayes (Questions 8-14)
8. Two-headed coin
You have 100 coins. 99 are fair, one has heads on both sides. You pick one at random and flip it 10 times. All 10 are heads. What is the probability you picked the two-headed coin?
Answer: ~91.2%. Bayes. (P(2H | 10H) = (1)(0.01) / [(1)(0.01) + (1/1024)(0.99)] = 0.01 / 0.01097 ≈ 0.912).
9. Disease test
A test for a disease has 99% sensitivity and 95% specificity. The disease has 1% prevalence. A person tests positive. What is the probability they have the disease?
Answer: ~16.7%. (P(D | +) = (0.99)(0.01) / [(0.99)(0.01) + (0.05)(0.99)] = 0.0099 / 0.0594 ≈ 0.167).
10. Three coin types
You have a bag with three coins: one fair, one always-heads, one always-tails. You pick one and flip it. It lands heads. What is the probability the next flip is also heads?
Answer: 5/6. Posterior: (P(2H|H) = (1/3)(1)/[(1/3)(0.5) + (1/3)(1) + (1/3)(0)] = (1/3)/(1/2) = 2/3). Posterior on fair: 1/3. Next-flip probability: ((2/3)(1) + (1/3)(0.5) = 5/6).
11. Boy or girl paradox
A family has two children. At least one is a boy. What is the probability both are boys?
Answer: 1/3. Sample space {BB, BG, GB, GG}, condition on "at least one boy" = {BB, BG, GB}. (P(BB) = 1/3).
12. Boy or girl paradox v2
A family has two children. At least one is a boy born on a Tuesday. What is the probability both are boys?
Answer: 13/27. Sample space (gender, day) pairs. Condition counts to 27 favourable, of which 13 have two boys. The day information makes the answer counter-intuitively close to 1/2.
13. Monty Hall
Three doors hide one car and two goats. You pick door 1. The host (knowing what's behind each door) opens door 3 to reveal a goat. Should you switch to door 2?
Answer: Yes. Switching wins with probability 2/3; staying wins with probability 1/3.
14. Spam filter
Your filter classifies 95% of spam emails correctly and 99% of legitimate emails correctly. 90% of all email is spam. An email is classified as legitimate. What's the probability it actually is?
Answer: ~63%. Bayes. (P(L | classified L) = (0.99)(0.1) / [(0.99)(0.1) + (0.05)(0.9)] = 0.099 / 0.144 ≈ 0.687). The poor specificity matters more here than sensitivity.
Section 3: Recursion and States (Questions 15-21)
15. Random walk hitting time
A symmetric random walk starts at 0. What is the expected number of steps to reach +N for the first time?
Answer: Infinite. A symmetric walk has infinite expected hitting time of any positive level. Be ready to discuss why - the walk is recurrent (eventually visits every level with probability 1) but the expected time is unbounded.
16. Drift random walk
A random walk moves +1 with probability 0.6 or -1 with probability 0.4. Expected steps to reach +10?
Answer: 50. With drift, expected hitting time of N is (N / (p - q) = 10/0.2 = 50).
17. Gambler's ruin
You have $5; you flip a fair coin and win $1 on heads, lose $1 on tails. You stop at $0 or $10. What is the probability you reach $10?
Answer: 0.5. For symmetric random walk between 0 and N, starting at k, P(reach N) = k/N = 5/10 = 0.5.
18. Pattern matching
You flip until you see HHT or TTH. What is the probability HHT appears first?
Answer: 0.5. By symmetry. Be careful - asymmetric pattern competitions often have non-obvious answers (HHH vs TTT is also 0.5; but HH vs TH is not).
19. Asymmetric patterns
Flip a fair coin until HH appears or TH appears. What is the probability HH appears first?
Answer: 1/4. TH is hard to "miss": once you see T, the next H gives TH. HH requires you to start a streak that doesn't get interrupted. Probability of HH first = (probability you start with H twice) = 1/4. After any T appears, TH wins on the next H, which is guaranteed to come before HH.
20. Coin until two consecutive different
Flip a fair coin until you see two consecutive different outcomes (HT or TH). Expected flips?
Answer: 3. First flip is anything (1 flip). After that, each subsequent flip is "different from the previous" with probability 0.5. Expected additional flips = 2. Total = 3.
21. Run length expectation
What is the expected length of the longest run of heads in 10 fair coin flips?
Answer: ~3.66. No clean closed form; computed via recursion. Worth knowing the rough answer (just under 4 for n = 10).
Section 4: Generating Functions and Variance (Questions 22-26)
22. Variance of product
X and Y are independent N(0,1). Compute Var(XY).
Answer: 1. (Var(XY) = E[(XY)^2] - (E[XY])^2 = E[X^2]E[Y^2] - 0 = 1).
23. Sum of dice
You roll two dice. Variance of the sum?
Answer: 70/12 ≈ 5.83. Each die has variance 35/12. Sum is 70/12.
24. Conditional expectation
(X, Y) iid N(0,1). (E[X | X+Y = z])?
Answer: z/2. By symmetry and the joint Gaussian property.
25. Negative binomial variance
What is the variance of the number of fair coin flips needed to see 10 heads?
Answer: 20. Negative binomial with p=0.5, r=10. Variance = (r(1-p)/p^2 = 10 \cdot 0.5 / 0.25 = 20).
26. Covariance of order statistics
Two iid uniform [0,1] random variables. Let (X_{(1)} < X_{(2)}) be order statistics. (Cov(X_{(1)}, X_{(2)}))?
Answer: 1/36. Standard result. (E[X_{(1)}] = 1/3), (E[X_{(2)}] = 2/3), and the joint density gives (E[X_{(1)} X_{(2)}] = 1/4). Covariance is (1/4 - 1/3 \cdot 2/3 = 1/4 - 2/9 = 1/36).
Section 5: Real Trading Interview Examples (Questions 27-30)
27. Optiver dice game
I roll five dice. I will pay you the median value. What's a fair price?
Answer: ~3.5. By symmetry of the uniform distribution. Worth knowing why and being ready to compute exactly: (E[X_{(3)}] = 3.5) for five uniform draws on {1..6}.
28. SIG card draw
You draw a card from a standard deck. If you draw a face card, you win $5. Otherwise you win the face value (Ace = 1). What's the expected value?
Answer: ~$4.92. Three face cards per suit (J, Q, K) = 12 face cards * $5 = $60. The 40 non-face cards have face values 1-10, each appearing 4 times. Sum = (4 \cdot (1+2+...+10) = 4 \cdot 55 = 220). Total expected payoff: ((60 + 220)/52 = 280/52 ≈ 5.38).
29. Jane Street ant on cube
An ant walks randomly on the edges of a cube. Starting at one corner, what is the expected number of steps to reach the opposite corner?
Answer: 10. Set up states by distance from the starting corner: distance 0, 1, 2, 3 (the antipode). Let (E_i) be the expected steps from distance i. By symmetry of the cube graph: (E_0 = 1 + E_1), (E_1 = 1 + (1/3)E_0 + (2/3)E_2), (E_2 = 1 + (2/3)E_1 + (1/3)E_3), (E_3 = 0). Solving: (E_0 = 10).
30. Citadel dollar auction
Two players alternate bidding for a $20 bill, in $1 increments. Loser pays their last bid; winner pays their last bid and gets the bill. What happens?
Answer: This is the dollar auction paradox. Game-theoretically, both players can rationally escalate well past $20. Real outcomes typically see bids of $40+ as players try to limit losses. The interviewer is testing whether you recognise this is a backward-induction trap, not a clean expected-value problem.
How to Use This Guide
Work through the questions in order, attempting each before reading the answer. Time yourself - the bar at top firms is roughly 60 to 90 seconds for a clean problem and 3 to 5 minutes for a multi-step recursion or Bayes problem.
For your overall preparation plan, see our quant interview questions hub. For specific firm interview content, see our SIG interview, Jane Street interview, Optiver interview and Citadel interview guides.
For more on probability theory itself, see our probability for quant finance and statistics for quantitative trading guides.
Practise the questions Quant Probability Interview Questions: 30 Worked Examples 2026 actually asks
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