Practise the reasoning, not the answers

You flip a fair coin three times. Given that at least one flip came up heads, what is the probability all three were heads?

Two people independently pick a real number uniformly from [0, 1]. What is the probability the two numbers differ by more than 0.25?

You draw cards one at a time without replacement from a standard 52-card deck until the first ace appears. What is the expected number of cards drawn?

A random walk starts at 0 and moves +1 or −1 with equal probability each step. What is the probability it reaches +3 before it reaches −5?

What is 17 × 23, without writing anything down?

Estimate the square root of 4,600 to one decimal place in under ten seconds.

A contract pays 100 if an event happens and 0 otherwise. It trades at 37. You think the true probability is 41%. You can trade 500 contracts. What is your expected edge?

I roll two fair six-sided dice and pay you the product. Make me a market, two wide.

You are quoting a market and get lifted five times in a row on the offer, in full size, by the same counterparty. Nothing else in the market has moved. What do you do?

Make a market on the number of petrol stations in the United Kingdom.

You backtest 200 signals and the best one has a Sharpe ratio of 2.4. Should you trade it?

Your alpha shows a strong in-sample relationship with next-day returns, but the effect halves out of sample. Give three explanations that are not overfitting.

Two return series each have a correlation of 0.6 with a third. What is the tightest bound you can put on their correlation with each other?

Design a data structure that supports insert, delete and get-random-element, all in expected O(1).

You receive a stream of order-book updates and must report the maximum trade price in the last N events at every step. How?

How would you implement a limit order book supporting add, cancel and match, and what is the complexity of each?

Why might std::unordered_map be a poor choice on a latency-critical hot path?

Your median tick-to-trade latency is fine but the 99.9th percentile is ten times worse. Name the likely causes.

Explain false sharing and how you would detect it in a multi-threaded market data handler.

Why is k-fold cross-validation usually wrong for financial time series, and what do you use instead?

A gradient-boosted model predicts next-hour returns with an out-of-sample R² of 0.004. Is that useful?

You have 25 horses and a track that races five at a time, with no timer. What is the minimum number of races to find the fastest three?

One hundred prisoners each open 50 of 100 boxes searching for their own number; all must succeed or all fail. What strategy beats a vanishingly small success rate?