Quant interview questions, with the reasoning written out

You roll a fair six-sided die until the running total first exceeds 12. What is the probability the final roll is a 6?

A family has two children. You are told at least one of them is a boy. What is the probability both are boys?

Three doors hide one prize. You pick a door. The host, who knows where the prize is, opens a different door and reveals it is empty, then offers you the swap. Do you swap?

You and I alternate flipping a fair coin, you first. The first person to flip heads wins. What is your probability of winning?

A stick of length one is broken at two points chosen uniformly and independently. What is the probability the three pieces form a triangle?

How many people do you need in a room before the probability that two share a birthday exceeds one half?

What is the expected number of fair coin flips until you see two heads in a row?

A symmetric random walk on the integers starts at zero. Does it return to zero with probability one, and what is the expected time to return?

Two envelopes contain money, one twice the other. You open one and find 100. Swapping appears to have expected value 125. What is wrong?

A condition affects one person in a thousand. A test detects it 99 percent of the time and has a 5 percent false positive rate. You test positive. What is the probability you have it?

N people leave hats at a cloakroom and receive them back in a uniformly random order. What is the expected number of people who get their own hat?

You roll a fair six-sided die and may either keep the value or reroll once and keep the second value. What is the optimal strategy and its expected value?

Two fair six-sided dice are rolled. What is the expected value of the larger?

You deal a shuffled 52-card deck one card at a time. What is the expected number of cards dealt before the first ace appears?

A cereal brand hides one of six equally likely toys per box. How many boxes until you expect to own all six?

You interview n candidates in random order, learn only relative ranks, and must accept or reject each immediately. How do you maximise the chance of hiring the best one?

You start with 10 and bet 1 per round on a fair coin, stopping at 0 or at 20. What is the probability you reach 20, and how long does the game last on average?

You draw independent uniform numbers on [0,1] and stop when their sum first exceeds 1. How many draws do you expect to make?

Two ropes each burn for exactly one hour but not at a uniform rate. Using only these, measure forty five minutes.

One hundred prisoners must each find their own number among one hundred boxes, opening at most fifty each, with no communication. All must succeed. What strategy beats 2^-100?

One of one thousand bottles is poisoned, and the poison acts in exactly one day. How many testers do you need to identify the bottle in a single day?

On an island everyone sees others' eye colour but not their own, and anyone who deduces theirs is blue must leave that night. A visitor says aloud 'at least one of you has blue eyes'. With one hundred blue-eyed islanders, what happens?

How many piano tuners work in New York City?

Twelve identical-looking balls, one of which differs in weight, and a balance scale. Find the odd ball and whether it is heavy or light in three weighings.

Ants are placed at random points on a one-metre pole, each facing a random direction, all walking at one metre per minute. When two meet they reverse. How long until the pole is clear?

Make me a market on the sum of two fair dice.

You quote a market on a coin flip paying 100 for heads. Your counterparty may have seen the coin. How should your quote change?

You have been lifted repeatedly and are now long well beyond your limit. What do you do with your quote?

What determines how wide you quote?

Make a market on the number of petrol stations in the country you are sitting in.

You have an edge: a bet wins 60 percent of the time at even money. What fraction of your capital do you stake?

A and B are 0.9 correlated, B and C are 0.9 correlated. What range is possible for the correlation of A and C?

A fund's best traders this year are usually worse next year. Does that mean the firm is failing to develop them?

A strategy shows a Sharpe of 2 over six months. How much should that convince you?

You tried two hundred signal variants and kept the best. Its backtest Sharpe is 2.5. What do you actually believe?

Why does stationarity matter before you model a price series, and what do you do about it?

A strategy outperforms in every market regime measured separately but underperforms overall. How?

Why does the sample variance divide by n-1 rather than n?

Given an array of prices in time order, compute the maximum drawdown in a single pass.

Maintain the median of the last k values of a stream efficiently.

What data structures would you use for a limit order book that must add, cancel and match quickly?

Pick one element uniformly at random from a stream whose length you do not know in advance, using constant memory.

Two implementations of matrix multiplication have identical complexity but one runs several times faster. Why?

How would you price a European call by Monte Carlo, and how do you make the error small enough to be useful?

Why should you not sum prices as floating point, and what do you do instead?