Quant interviews are a different beast from standard software engineering interviews. Firms like Jane Street, Citadel, Two Sigma, DE Shaw, and HRT are testing your ability to think mathematically under pressure. The questions cover probability, expected value, combinatorics, and brain teasers that require creative reasoning.
This guide covers the core math topics, the most common question types, and strategies for working through problems you have never seen before.
What quant firms are actually testing
The goal is not to see if you have memorized formulas. They want to see:
- How you think through problems - Can you break a complex problem into simpler parts?
- Whether you can handle ambiguity - Many problems are intentionally vague. Can you ask the right clarifying questions?
- Speed and accuracy - Mental math matters. Being able to estimate quickly and catch your own errors is a signal they watch for.
- Your comfort with uncertainty - Probability is the language of finance. If you freeze up on conditional probability questions, that is a red flag.
Probability fundamentals
Probability is the single most important topic. If you are short on time, study this section and skip everything else.
Conditional probability and Bayes' theorem
P(A|B) = P(B|A) * P(A) / P(B). This formula comes up constantly, both directly and in disguise. Make sure you can apply it to word problems without hesitation.
Classic example: A test for a disease is 99% accurate (sensitivity and specificity). The disease affects 1% of the population. If someone tests positive, what is the probability they actually have the disease?
Answer: P(disease|positive) = (0.99 * 0.01) / (0.99 * 0.01 + 0.01 * 0.99) = 0.5. This surprises most people. The key insight is that when the base rate is low, even a good test produces many false positives.
Expected value
Expected value (EV) is the weighted average of all possible outcomes. E[X] = sum of (value * probability) for each outcome. Many interview problems reduce to an EV calculation.
Key property: Linearity of expectation. E[X + Y] = E[X] + E[Y], even when X and Y are not independent. This is one of the most powerful tools in probability and shows up in interview problems constantly.
Example: You roll a fair die. What is the expected number of rolls to see all 6 faces? Using linearity of expectation: the expected rolls for the kth new face is 6/(6-k+1). So the total is 6/6 + 6/5 + 6/4 + 6/3 + 6/2 + 6/1 = 14.7 rolls. This is the "coupon collector problem."
Combinatorics
You need to be comfortable with permutations, combinations, and the multiplication principle. The main formulas:
- Permutations: n! / (n-k)!
- Combinations: n! / (k!(n-k)!)
- Stars and bars: distributing n identical items into k bins = C(n+k-1, k-1)
Don't just memorize these. Understand when each one applies. If order matters, use permutations. If it doesn't, use combinations.
Brain teasers and logic puzzles
These are designed to test creative thinking. There is no formula to apply. The key strategies:
- Start with small cases. If the problem involves n objects, try n=1, n=2, n=3 and look for a pattern.
- Work backwards. Sometimes the problem is much easier when you start from the end state.
- Look for symmetry. Many teasers have elegant solutions that exploit symmetry in the problem.
- Use expected value. When a problem involves randomness, computing EV often simplifies things dramatically.
Classic brain teasers
The 100 prisoners and light switch: 100 prisoners are in solitary cells. There is a room with a light switch. Prisoners are randomly brought to the room one at a time. They must devise a strategy (agreed upon beforehand) so that one prisoner can eventually declare with certainty that all 100 have visited the room. The strategy: designate one "counter" who is the only one allowed to turn the light off. Everyone else turns the light on the first time they find it off. When the counter has turned it off 99 times, everyone has visited.
Two envelopes problem: You are given two envelopes. One contains twice as much money as the other. You pick one and see $100. Should you switch? The naive expected value argument suggests always switching, but this leads to a paradox. The resolution involves understanding that you can't assume a uniform prior over all possible amounts.
Breaking a chocolate bar: You have a 4x6 chocolate bar (24 squares). How many breaks do you need to separate it into individual squares? Answer: 23. Each break increases the number of pieces by exactly one. You start with 1 piece and need 24, so you need 23 breaks. This generalizes to m*n - 1 for any rectangular bar.
Market making and estimation
Some quant interviews include market-making games or Fermi estimation questions. These test your ability to think about uncertainty quantitatively.
Market making basics
A market maker provides a bid (price to buy) and an ask (price to sell) for some quantity. Your profit is the spread between bid and ask. The challenge is setting prices that are tight enough to attract trades but wide enough to protect you from adverse selection (trading against someone who knows more than you).
In interview games, you'll be asked to make markets on things like "how many windows are in this building" or "what will the temperature be tomorrow." The key is to:
- Start with your best estimate as the midpoint
- Set a spread that reflects your uncertainty
- Update your estimate as you receive new information
- Widen spreads when you're less confident, tighten when you're more confident
Fermi estimation
Fermi questions ask you to estimate a quantity with no obvious way to look it up. "How many piano tuners are in Chicago?" "How many golf balls fit in a school bus?" The trick is to break the problem into smaller estimates that you can reason about.
For the piano tuners question: estimate the population of Chicago (~3M), the fraction of households with a piano (~5%), how often pianos need tuning (~1x/year), how many pianos a tuner can service per day (~4), and how many working days per year (~250). That gives you roughly 3M * 0.05 / (4 * 250) = 150 piano tuners. The exact number does not matter. They want to see your decomposition.
Mental math tips
Speed on mental math genuinely matters in quant interviews. A few things worth practicing:
- Multiply by breaking apart. 17 * 23 = 17 * 20 + 17 * 3 = 340 + 51 = 391.
- Know common fractions. 1/7 = 0.143, 1/8 = 0.125, 1/9 = 0.111. These come up when computing probabilities.
- Estimate with powers of 2. 2^10 = 1024, which is roughly 1000. This helps with quick order-of-magnitude estimates.
- Sanity check your answers. If you calculate a probability greater than 1 or a negative expected value for something that is always positive, you made an error. Catch it before the interviewer does.
Distributions you should know
- Bernoulli/Binomial - Number of successes in n independent yes/no trials. Mean = np, variance = np(1-p).
- Geometric - Number of trials until the first success. Mean = 1/p.
- Poisson - Number of events in a fixed interval when events occur at a constant rate. Mean = variance = lambda.
- Normal - The bell curve. Know the 68-95-99.7 rule (percentage of data within 1, 2, 3 standard deviations).
- Uniform - All outcomes equally likely. Mean = (a+b)/2, variance = (b-a)^2/12.
How to prepare
- Do probability problems daily. Even 20 minutes a day builds the intuition you need. Textbooks like Blitzstein's "Introduction to Probability" or the "Fifty Challenging Problems in Probability" by Mosteller are solid resources.
- Practice mental math. Use flashcards or apps. Time yourself on multiplication, division, and percentage calculations.
- Play estimation games. With friends, take turns making markets on random questions. This builds the right intuition for handling uncertainty.
- Study past questions from your target firms. Jane Street, Citadel, and Two Sigma each have their own style. Knowing what to expect reduces anxiety and lets you focus on solving.