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Probability and expected value questions in quant interviews
Interviews for trading and quant roles ask short probability questions and expect an exact answer said out loud: a fraction, with the reasoning behind it. The questions use dice, cards, coins and simple games. The maths is school level; what is tested is whether you set the problem up correctly and avoid a handful of standard traps.
The kinds of question
- Counting outcomes: "What is the probability that two dice add up to 10 or more?" Six of the 36 rolls do (4-6, 5-5, 6-4, 5-6, 6-5, 6-6), so 1/6.
- At least one: "What is the probability of at least one six in four rolls?" Work out the chance of none and subtract: 1 − (5/6)⁴ = 671/1296, about 52%.
- Without replacement: two aces from a deck is 4/52 × 3/51, because the deck shrinks.
- Conditional probability: "At least one of two coins is heads. What is the chance both are?" Three outcomes fit what you were told, so 1/3.
- Bayes: "2% of people have a condition. A test flags 90% of those who have it and 20% of those who don't. Someone tests positive: how likely is it they have it?" Out of 10,000 people, 180 true positives and 1,960 false ones: 180 ÷ 2,140 = 9/107, about 8%.
- Expected value: "What is a fair price for a game that pays the number you roll?" The average roll, 3.5.
- Games with a choice: "You may reroll once. What is the game worth?" Keep a 4, 5 or 6, reroll otherwise: 4.25.
The traps
- Adding when cases overlap. Four rolls do not give a 4/6 chance of a six. If your total could pass 100%, you have added overlapping cases.
- Treating unequal outcomes as equal. Two dice have 36 ordered outcomes; the 11 possible sums are not equally likely.
- Confusing P(A given B) with P(B given A). A test that catches 90% of cases does not make a positive result 90% likely to be right.
- Forgetting the stake. Profit is the expected payout minus what you paid to play.
- Squaring the average. A die that pays its square is worth 91/6, about 15.2, not 3.5² = 12.25.
The quant interview maths guide covers each of these, with one-line shortcuts for the common questions.
How to prepare
- Say the set-up first: the outcomes you are counting, or the step that repeats. Interviewers mark the reasoning as much as the number.
- Give the exact fraction, then a rough decimal: "91/6, a bit over 15".
- Check the answer makes sense: a probability is between 0 and 1, and an expected value lies between the smallest and largest payout.
- Practise without options. An interviewer gives you none to choose from, so at the top levels here you type the fraction yourself.
Start now
Probability and expected value practice here is free up to level 3, and every question explains its answer step by step. Levels 4 to 6, where the conditional, Bayes and game questions get harder, come with paid access; at the top two you type the answer. There are drills for counting, statistics and mental arithmetic too.