Quant interview questions, by role

A fair coin is tossed twice. Given that at least one toss is heads, what is the probability that both are heads?

What is the expected number of flips of a fair coin until the first head appears?

Two fair six-sided dice are rolled. What is the expected value of the larger of the two (ties count as that value)?

A disease affects 1% of a population. A test is 99% sensitive (true-positive) and has a 1% false-positive rate. Given a positive test, what is the probability the person actually has the disease?

What is the smallest number of people in a room for which the probability that at least two share a birthday exceeds 50%? (Ignore leap years.)

You roll a fair six-sided die repeatedly. What is the expected number of rolls until you have seen all six faces?

Three doors: one hides a car, two hide goats. You pick a door; the host (who knows) opens a different door revealing a goat, then offers you the switch. What is your probability of winning the car if you switch?

Five rational pirates split 100 gold coins. The most senior proposes a split; if at least half (including the proposer) approve, it passes — otherwise he is thrown overboard and the next-senior proposes. Pirates prioritise survival, then gold, then bloodthirst. How many coins does the most senior pirate keep?

You have 2 identical eggs and a 100-floor building. An egg breaks if dropped from floor F or above. Minimise the number of drops in the worst case needed to find F with certainty. How many drops?

100 lockers start closed. Person i (for i = 1…100) toggles every locker that is a multiple of i. After all 100 people pass, how many lockers are open?

12 identical-looking balls include exactly one of a different weight — you don’t know if it’s heavier or lighter. Using only a balance scale, what is the minimum number of weighings that guarantees finding the odd ball AND whether it is heavy or light?

You have 1000 bottles of wine, exactly one of which is poisoned. A test strip reacts (after one hour) if it touches any poison. With only one hour before a feast, what is the minimum number of test strips needed to guarantee finding the poisoned bottle?

An interviewer asks you to make a market on the sum of two fair six-sided dice. Before you quote a bid and an ask, you need the fair value. What is it?

You roll a fair die and are paid its face value in dollars, but you may re-roll up to two times, keeping only the final roll. Playing optimally, what is this game worth?

A binary asset settles at 100 or 0, each with probability 1/2. Of the traders who arrive, 20% are informed (they know the outcome and trade in its direction) and 80% are noise traders who buy or sell with probability 1/2 each. A single trader arrives and buys from you at your ask. To break even in expectation against this flow, what is the lowest ask you can quote?

A repeatable bet wins with probability 40%. A win pays 2-to-1 (win $2 per$1 staked); a loss costs the stake. What fraction of your bankroll should you stake per bet to maximise long-run growth (the Kelly fraction), in percent?

Everyone in a large group simultaneously picks a number from 0 to 100. The winner is whoever is closest to two-thirds of the group average. If every player is perfectly rational and knows everyone else is too, what number does game theory say everyone picks?

Cards from a shuffled standard 52-card deck are revealed one at a time. Your running total gains $1 for every red card revealed and loses$1 for every black card. You may stop whenever you like and collect the running total (you may also stop immediately at $0). Playing optimally, what is this game worth?

You start with $30 and bet$1 on a fair coin each round, stopping when you reach $0 or$100. What is the probability you reach $100 before going broke?

What is the expected number of fair-coin flips to get two heads in a row (HH)?

For a standard Brownian motion $B_t$, what is $\mathbb{E}[B_3^2]$?

European call and put share strike K = 100 on a non-dividend stock at S = 100, with rate r = 0 and maturity T = 1. The call trades at 8. What is the fair price of the put?

A stock at 100 will move to 120 or 80 in one period. With interest rate r = 0, what is the risk-neutral probability of an up move?

Roughly what is the delta of an at-the-money European call on a non-dividend stock with short time to expiry?

A stock follows geometric Brownian motion $dS_t = \mu S_t\,dt + \sigma S_t\,dW_t$ with $S_0$ given. What is $\mathbb{E}[S_T]$?

State the Central Limit Theorem, and give the standard deviation of the sample mean of n i.i.d. draws with variance σ².

In a simple linear regression of Y on X, the slope is Cov(X, Y)/Var(X). If Cov(X, Y) = 4 and Var(X) = 8, what is the slope?

Tanks are serial-numbered 1, 2, …, N. You capture 4 tanks and the largest serial number observed is 60. Using the standard minimum-variance unbiased estimator, what is your estimate of N?

X and Y are independent random variables with the same variance. What is the correlation between X and X + Y?

You average 100 iid observations, each with variance 25. What is the variance of the sample mean?

A drug shows a higher recovery rate than placebo among men, and a higher recovery rate than placebo among women — yet a lower recovery rate overall when the groups are pooled. Explain precisely how this can happen, and give a concrete numeric example.

A portfolio returns 12% with 20% volatility; the risk-free rate is 2%. What is its Sharpe ratio?

A portfolio’s daily return is Normal with mean 0 and standard deviation 2%. What is the 1-day 95% Value at Risk, expressed as a percent of portfolio value? (e.g. 3.29)

Assuming i.i.d. normal daily returns, by what factor do you scale a 1-day VaR to get a 10-day VaR? (Round to 2 decimals.)

Two uncorrelated assets have volatilities 10% and 20%. In the minimum-variance portfolio, what weight goes to the 10% (lower-vol) asset?

You can repeatedly make an even-money bet (win or lose the amount staked) that you win with probability 60%. What fraction of your bankroll should you bet each time to maximize long-run growth?

A fund’s value series is 100, 120, 90, 110, 80, 130. What is the maximum drawdown, as a percent? (Round to 2 decimals.)

You throw N uniform random points into the unit square and count the fraction that land inside the quarter unit circle. To estimate π, what should you multiply that fraction by?