Quant probability interview questions: how to reason out loud
In a quant probability interview the number matters, but the route to it is what gets scored, and the interviewer can only score the route you say. Restate the problem, name the tool you are reaching for (symmetry, conditioning on the first step, linearity of expectation), say each intermediate result, and check the answer against a bound or an extreme case before you commit.
Key points
- Restate the problem and its assumptions in one sentence before computing anything.
- Name the idea first: “by symmetry”, “condition on the first roll”, “linearity of expectation”.
- Say intermediate numbers so a slip can be caught early, by you or the interviewer.
- Check the answer: is it between 0 and 1, does an extreme case agree, is it near a quick estimate?
- Under time pressure, a clean approximate answer with its error stated beats silence.
A five-step spoken routine
- Restate: “Two fair ten-sided dice, I want the chance the first is strictly higher.”
- Assumptions: fairness, independence, what counts as a tie or a success.
- The idea: say which tool and why it fits before you calculate.
- The calculation, out loud, one intermediate result at a time.
- The check: a bound, an extreme case, or a rough estimate that should land nearby.
Four worked problems
Symmetry. Two people each roll a fair ten-sided die numbered 1 to 10. What is the chance the first roll is strictly higher? A tie happens with probability 10 / 100 = 1/10. By symmetry, “first higher” and “second higher” are equally likely, so each is (1 − 1/10) / 2 = 0.45. Check: 0.45 + 0.45 + 0.10 = 1.
Conditioning on the first step. A part passes inspection with probability 0.8 on each independent attempt, and you need two passes in a row. Let E₀ be the expected attempts from scratch and E₁ after one pass. Then E₁ = 1 + 0.2 × E₀, because a failure sends you back to the start, and E₀ = 1 + 0.8 × E₁ + 0.2 × E₀. Substituting gives E₀ = 1.8 + 0.36 × E₀, so E₀ = 1.8 / 0.64 = 2.8125. Check: at least 2 attempts are needed, and with a pass rate near 1 the answer should approach 2. It does.
Bayes. A signal fires on 90% of days when a stock gaps at the open and on 5% of other days. Gaps happen on 4% of days. Given the signal fired, the chance of a gap is 0.04 × 0.9 / (0.04 × 0.9 + 0.96 × 0.05) = 0.036 / 0.084 = 3/7, about 43%. Check: the base rate is low, so a fairly accurate signal still leaves less than an even chance.
Expected value with a choice. Roll a fair eight-sided die; you may roll once more and must keep the second result. You are paid the final face. The second roll is worth 4.5 on average, so keep any first roll above 4.5, which means 5 to 8. Value: (5 + 6 + 7 + 8) / 8 + (4/8) × 4.5 = 3.25 + 2.25 = 5.5. Check: it must lie between 4.5 (never reroll) and 8.
How it sounds out loud
“I’ll set up states: zero passes in a row, or one. E-zero and E-one for the expected attempts from each.”
“From one pass, I spend an attempt; 80% of the time I’m done, 20% I’m back to zero. So E-one is 1 plus 0.2 E-zero.”
“Plugging in, E-zero is 1.8 over 0.64, which is 2.8125.”
“Sanity check: never below 2, and if the pass rate were 1 it would be exactly 2. Fine.”
Two more tools worth saying by name
Linearity of expectation. Twelve people drop their badges in a box and each draws one at random. How many people, on average, get their own badge back? Each person has a 1/12 chance, and the expected count is the sum of those chances whether or not the events are independent: 12 × 1/12 = 1. Saying “linearity, so I don’t need the joint distribution” is the whole answer, and it signals that you know why the shortcut is allowed.
Counting in sequence. Draw three cards from a standard 52-card deck without replacement. What is the chance all three are different suits? The first card can be anything. The second must avoid the first suit: 39 of the remaining 51. The third must avoid both: 26 of the remaining 50. So 39/51 × 26/50 = 1,014 / 2,550, about 0.40. Check: three cards from four suits, so “all different” should be common but not certain, and 40% fits.
When a problem looks like it needs a long case analysis, pause and say which of these tools might shorten it. Interviewers often extend a problem in exactly the direction where the shortcut matters, such as “now do it for n people”.
Speed and estimation
Most rounds put a clock on each problem. When the exact answer is slow, say the estimate and how far off it could be: “3 / 7 is a bit under a half, about 0.43.” Keep a few reference values in your head: 1/7 ≈ 0.143, 1/e ≈ 0.368, the square root of 2 ≈ 1.414, and (1 − 1/n)ⁿ near 1/e for large n.
If you notice a mistake, say so and fix it out loud. Prep guides report that catching your own error counts in your favor; hiding it does not.
Common mistakes
- Computing before restating the problem, and solving a slightly different one.
- Silent work on paper followed by a bare number.
- Assuming independence without saying so, or where it fails.
- Missing the tie case in “higher than” questions.
- No sanity check, so an answer above 1 or below a hard bound slips through.
- Freezing on an exact form when a stated approximation would do.
Questions
Which topics come up in quant probability interviews?
Expected value and linearity of expectation, conditional probability and Bayes, combinatorics, Markov-chain style “first step” problems, simple games with a choice, and quick estimation.
Should I use paper?
Paper is fine for arithmetic, but keep talking while you write. The interviewer can only score the reasoning they hear.
What if I get the wrong number with the right method?
A clear method with an arithmetic slip that you check and fix usually scores well. A right number with no visible reasoning often does not.
Practise this out loud with Vera
Mockich’s probability round runs 30 min: five short problems on expected value, combinatorics, conditional probability, estimation and market making, reasoned out loud. Once you answer, Vera asks you to sanity-check it or extends the problem once, and gives at most two small nudges. Market-making questions are asked like the others; there is no live trading game. The report scores correctness, structured reasoning, speed and estimation, communication (plus market intuition at lead) and quotes what you said.
Related guides
Sources
We read these sources on 2026-11-03 for the problem types and the habits interviewers reward; the four problems, their numbers and the spoken lines are our own.
- [1] Ricardo Vieira, Probability puzzles (notebook) (MIT)
- [2] dwcoder on GitHub, An Interview Primer for Quantitative Finance (CC BY-SA 4.0)
- [3] techinterview.org, Mental math in trading interviews
- [4] TraderMath, Quant interview questions