A cheat sheet will not get you through a quant interview — but walking in without one memorized will get you thrown out of it. Interviewers at trading firms assume the standard results are instant recall: linearity of expectation, geometric waiting times, put-call parity, gambler's ruin. The interview then tests whether you can combine them under time pressure. This page is the working version of that sheet; a condensed one-page reference lives on our formula cheat sheet page if you want something printable for the night before.
Why interviewers assume you have this memorized
A first-round quant screen is typically 30–45 minutes and covers 3–5 problems. If you spend four of those minutes re-deriving $E[X] = 1/p$ for a geometric random variable, you have burned a problem's worth of time on something the interviewer considers table stakes. The firms are explicit about this in how they grade: fluency with the base results signals practice volume, and practice volume predicts how you will handle the follow-up chain — which is where offers are actually decided. The base results are a floor, not the test.
Probability & expectation: the core block
These cover the majority of what shows up in probability interview questions at every tier of firm.
| Result | Formula | Where it shows up |
|---|---|---|
| Linearity of expectation | $E[X+Y]=E[X]+E[Y]$, no independence needed | Indicator-variable tricks, matching problems, dice sums |
| Geometric waiting time | $E[\text{trials until success}]=1/p$ | First heads, first six, coupon-collector pieces |
| Bayes' theorem | $P(A\mid B)=\dfrac{P(B\mid A)P(A)}{P(B)}$ | Biased-coin identification, false-positive tests |
| Variance identities | $\mathrm{Var}(X)=E[X^2]-E[X]^2$; variances add for independent $X, Y$ | Sums of dice, portfolio-flavored follow-ups |
| Gambler's ruin (fair game) | Starting at $a$, target $a+b$: $P(\text{win})=\dfrac{a}{a+b}$ | Random walks, betting sequences |
| Kelly fraction | $f^{*}=p-\dfrac{q}{b}$ for a $b$-to-1 payoff | Sizing questions after any betting-game problem |
The single highest-leverage item is linearity of expectation with indicator variables. A large fraction of "hard" expectation problems collapse to summing $n$ easy probabilities once you define the right indicators.
Markov chains & stopping problems
The workhorse technique is first-step analysis: define the expected value from each state, write one equation per state by conditioning on the next step, and solve the linear system. It handles expected hitting times, absorption probabilities, and most pattern-waiting problems. If states and transitions feel shaky, the Markov chain interview guide builds this up from scratch. Two facts worth having cold: for an absorbing chain, absorption probabilities also come from first-step equations, and for a biased random walk the ruin probability picks up a $(q/p)^a$ ratio structure instead of the simple fair-game fraction.
Options & linear algebra in one pass
For trading-desk interviews the options block is short but non-negotiable: put-call parity $C - P = S - Ke^{-rT}$, the sign and shape of the main Greeks (delta in $[0,1]$ for calls, gamma and vega peak near the money, theta usually negative for long options), and the no-arbitrage bounds on option prices. Interviewers rarely ask you to reproduce the Black-Scholes derivation; they ask you to reason about how the Greeks move when spot or vol moves — the options pricing question bank is built around exactly those follow-ups. For researcher roles, add the linear algebra minimum: trace equals the sum of eigenvalues, determinant equals their product, symmetric matrices have real eigenvalues and orthogonal eigenvectors, and a covariance matrix must be positive semidefinite. Our linear algebra questions drill each of these as they actually get asked.
Worked example: expected flips to see HH
Question: A fair coin is flipped until two heads appear in a row. What is the expected number of flips?
This is first-step analysis straight off the sheet. Define $E_0$ as the expected remaining flips with no progress, and $E_1$ as the expected remaining flips when the last flip was a head. Condition on the next flip:
$$E_0 = 1 + \tfrac{1}{2}E_1 + \tfrac{1}{2}E_0, \qquad E_1 = 1 + \tfrac{1}{2}\cdot 0 + \tfrac{1}{2}E_0.$$
The first equation gives $E_0 = 2 + E_1$. Substituting the second, $E_0 = 2 + 1 + \tfrac{1}{2}E_0$, so $\tfrac{1}{2}E_0 = 3$ and $E_0 = 6$.
The standard follow-up: expected flips to see HT is $4$, not $6$ — after a head, a tail finishes you, but after HH-progress a tail resets you to zero. If you can explain why the two patterns differ, you have demonstrated exactly the thing the memorized sheet cannot: understanding of the state structure.
The traps: where a memorized sheet backfires
- Applying $1/p$ to overlapping patterns. Waiting times for multi-flip patterns depend on self-overlap (HH vs HT above). The geometric formula only covers single-trial success.
- Using the fair gambler's ruin formula on a biased game. Interviewers deliberately set $p \neq 1/2$ to catch autopilot answers.
- Adding variances of dependent variables. $\mathrm{Var}(X+Y)$ needs the covariance term unless independence is stated.
- Quoting put-call parity without the discount factor. $C-P = S-K$ is wrong for any nonzero rate and interviewers notice.
- Reciting instead of reasoning. Every result above will draw a "why?" follow-up. Memorize the formula and the two-line argument behind it.
The sheet only pays off once it is automatic under a clock. Drill the results against real problems in the QuantVault problem bank, grab the downloadable set on the quant interview questions PDF page, and pressure-test your recall in our trading games — the same formulas, with money and seconds on the line.
More topic guides
- The Airplane Seat Problem: Why the Answer Is 1/2 (Three Proofs)
- Bayes' Theorem in Quant Interviews
- Behavioral Interview Questions at Trading Firms (With Answer Frameworks)
- Coin Flip Questions in Quant Interviews
- C++ Low-Latency Interview Questions at HFT Firms
- Dice Questions in Quant Interviews
- Fermi Estimation Interview Questions at Trading Firms
- Gambler's Ruin in Quant Interviews
- The Kelly Criterion in Quant Interviews
- Linear Regression Interview Questions: OLS Assumptions, R² Traps & Regression to the Mean
- Machine Learning Quant Interview Questions: Overfitting, Cross-Validation & Feature Leakage
- The Market Making Game Interview: How to Answer 'Make Me a Market'
- Market Microstructure Interview Questions: Order Books, Spreads & Adverse Selection
- Markov Chains in Quant Interviews
- Martingales in Quant Interviews
- Mental Math for Trading Interviews: Training Plan, Zetamac Benchmarks & Firm Tests
- The Monty Hall Problem — and the Variants Interviews Actually Ask
- Number Sequence Tests in Trading Interviews: The 8 Pattern Types & How to Practice
- Optimal Stopping in Quant Interviews
- Options Pricing Interview Questions: Black-Scholes, Greeks & Put-Call Parity
- Random Walks in Quant Interviews
- Statistics Interview Questions for Quant Roles: Hypothesis Testing, MLE & p-Value Traps
- Stochastic Calculus Interview Questions: Ito's Lemma, SDEs & Brownian Motion
- Time Series Interview Questions: Stationarity, ARMA & Autocorrelation Traps
- Top 50 Quant Interview Questions (With Full Solutions)
- All guides & explainers
Frequently asked questions
What should a quant interview cheat sheet include?
The core blocks are probability and expectation (linearity of expectation, geometric waiting times, Bayes' theorem, variance identities), Markov chain first-step analysis, gambler's ruin, the Kelly fraction, put-call parity with the Greeks' signs and shapes, and basic eigenvalue facts for linear algebra. That set covers the standard results behind most first-round quant questions. The sheet should also record the two-line argument behind each formula, since interviewers almost always ask why a result holds.
Is memorizing a cheat sheet enough to pass a quant interview?
No. Interviewers treat the standard formulas as a floor and test whether you can combine them under follow-up pressure, often by tweaking a parameter so the memorized version no longer applies (a biased coin, an overlapping pattern, dependent variables). Memorization buys you speed on the easy steps; solving the follow-up chain is what actually earns the offer.
What is the expected number of coin flips to get two heads in a row?
For a fair coin, the expected number of flips to see HH is 6. It follows from first-step analysis: writing expected-remaining-flips equations for the no-progress state and the one-head state and solving the two-equation linear system. The related pattern HT takes only 4 flips on average, because a failure after partial HT progress does not reset you to zero the way it does for HH.
Why is the answer for HH different from HT if both patterns have probability 1/4?
The difference is self-overlap. When you are waiting for HH and the second flip comes up tails, you lose all progress and restart; when you are waiting for HT and the second flip comes up heads, that head still counts as the start of a new attempt. Less overlap means failures waste less work, so HT arrives sooner on average (4 flips) than HH (6 flips).
Practice the real thing
QuantVault has 2,800+ quant interview problems with full solutions, intuition, and hints, firm-by-firm interview funnels, and an auto-graded coding judge. Start free.