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Many quant trading interviews ask you to "make a market": name a price you would buy at and a price you would sell at, for something nobody knows yet. This free drill trains the first half of that skill: finding the exact fair value fast. No account, and nothing you type leaves your browser.
The game needs JavaScript. The scoring rule, a worked round and every formula it uses are below, and they work without it.
10 rounds, 30 seconds each. Each round names a number made of fair dice, coins or cards. Type a bid (the price you would buy at) and an ask (the price you would sell at). You score when the fair value is inside your market, and more when the market is tight.
Why.
Your score: of
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The fair value is the exact expected value of the number. The standard deviation (SD) says how far the number usually lands from it.
The best possible set scores 1000. A market 4 SDs wide or wider scores 0 even when it holds the fair value: it is too wide to be useful.
The number: the sum of 3 fair dice.
One die averages 3.5, so three average 10.5. One die has variance 35/12, and the variances of independent dice add, so the SD is √(3 × 35/12) ≈ 2.96.
Quote 9 to 12. The fair value, 10.5, is inside. The width is 3, which is 1.01 SDs. Score: 100 × (1 − 3 ÷ (4 × 2.96)) ≈ 75.
Quote 11 to 13 instead and the fair value is below your bid. A trader who knows it sells to you at 11 and gains 0.5 on average each time. Score: 0.
A market has two numbers, and each one says something. The middle says what you think the thing is worth. The width says how sure you are. Too wide, and nobody trades with you. Too tight around the wrong value, and the people who know more than you trade with you and you lose. That second risk is called adverse selection.
The question turns up in trading interviews in the US, the UK, continental Europe and India, from campus tests to final rounds.
In a real interview the game usually goes further: the interviewer trades with you, or gives you a new fact, and you move your market. This drill does not do that part. It trains the step before it: a fast, exact fair value. Every fair value here can be worked out exactly, so the tightest market that still holds it scores best. The drill does not test how wide to go when you cannot work the value out; the standard deviation it shows is there to tell you how far the number usually lands.
Three habits help in every round. Break the number into simple parts and add their means (linearity of expectation works even when the parts depend on each other). Add variances only when the parts are independent. And say your reasoning out loud: the interviewer is grading it as much as the number.
The game deals from these 16 kinds of question, with different sizes (2 to 5 dice, 10 to 100 flips, and so on). Each set of 10 rounds uses 10 different kinds.
| Question | Fair value (mean) | Standard deviation |
|---|---|---|
| Sum of n fair dice | 3.5 n | √(35n/12) |
| Largest of n fair dice | Σ P(max ≥ k), k = 1 to 6 | from P(max ≤ k) = (k/6)ⁿ |
| Product of two fair dice | 3.5 × 3.5 = 12.25 | √((91/6)² − 12.25²) ≈ 8.94 |
| Gap between two fair dice | 70/36 ≈ 1.94 | ≈ 1.43 |
| Different faces in n rolls | 6 (1 − (5/6)ⁿ) | from indicator pairs |
| Heads in n fair coin flips | n/2 | √n / 2 |
| Sixes in n rolls | n/6 | √(5n/36) |
| Rolls until the first six | 1/p = 6 | √((1−p)/p²) = √30 |
| Flips until the first head | 1/p = 2 | √2 |
| Flips until two heads in a row | 6 | √22 |
| Aces in a five-card hand | 5 × 4/52 = 5/13 | hypergeometric |
| Spades in a five-card hand | 5 × 13/52 = 1.25 | hypergeometric |
| Hearts in a thirteen-card hand | 13 × 13/52 = 3.25 | hypergeometric |
| Cards left in their place after a full shuffle | n × 1/n = 1 | 1 |
| Runs in n fair coin flips | 1 + (n−1)/2 | √(n−1) / 2 |
| Cards between the two jokers in a 54-card deck | 52/3 | ≈ 12.61 |
Free worked problems on this site that use the same ideas:
For the full plan, with question sets by type and a six-week study plan, see quant interview prep.
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