DRW interview questions

Trading intern · Interview · Expected value · reported by a candidate

You flip a fair coin 100 times.

1. What is the expected value of (number of heads) × (number of tails)?
   Answer and worked solution

   Answer: 2475

   Key idea. Write tails as 100 − H and expand.

   1. E[H(100 − H)] = 100E[H] − E[H²].
   2. E[H] = 50; E[H²] = Var(H) + 50² = 25 + 2500 = 2525.
   3. 5000 − 2525 = 2,475.

   Check. Just below 50 × 50 = 2500: any spread away from 50–50 lowers the product.

Interviewers may push further: What is P(H × T = 2,500)?

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Quant trader intern · Interview · Expected value · reported by a candidate

Repeatedly take the top card of a 52-card deck and put it back at a uniformly random position.

We assume the top card goes back into one of the 52 possible positions (including the top) uniformly at random.

1. What is the expected number of moves until the card that started at the bottom reaches the top?
   Answer and worked solution

   Answer: 234.98

   Key idea. Track the number of cards below the original bottom card: it only ever goes up.

   1. With k cards below it, a moved card lands below it with probability (k + 1)/52.
   2. So that step takes 52/(k + 1) moves on average.
   3. It reaches the top when 51 cards are below it: Σ_k=0⁵⁰ 52/(k + 1) = 52 × H₅₁.
   4. H₅₁ ≈ 4.519, so ≈ 235 moves.

   Check. One more move after that, and the deck is perfectly shuffled.

Interviewers may push further: After one more move from there, why is the deck perfectly shuffled?

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Trader Assistant · Interview · Combinatorics · reported by a candidate

Each face of a cube is painted a different one of six colours.

1. How many different cubes can you make, counting cubes that can be rotated into each other as the same?
   Answer and worked solution

   Answer: 30

   Key idea. Count colourings, then divide by the number of rotations.

   1. 6! = 720 colourings.
   2. A cube has 24 rotations (6 faces on top × 4 turns), and with all colours different each gives a different colouring.
   3. 720/24 = 30.

   Check. Directly: colour 1 on the bottom, 5 choices for the top, then (4 − 1)! = 6 circular orders: 5 × 6 = 30.

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Quant Trading Analyst intern · Interview · Markov chains · reported by a candidate

A rook starts in the bottom-left corner of an empty chessboard and moves at random.

We assume each move goes to one of the 14 squares a rook can reach from where it stands, uniformly at random.

1. What is the expected number of moves until it reaches the top-right corner?
   Answer and worked solution

   Answer: 70

   Key idea. Collapse the 63 squares into 3 states: "neither in line with the target", "in line" (same row or column), and the target.

   1. From "neither": 1 of the 7 row moves and 1 of the 7 column moves line up with the target: P = 2/14 = 1/7 to "in line".
   2. From "in line" (say same column): 1 move hits the target, 6 stay in that column, 7 leave it.
   3. N = 1 + (1/7)A + (6/7)N gives N = 7 + A. A = 1 + (6/14)A + (7/14)N gives A = 63.
   4. From the corner (not in line): N = 70.

   Check. A random square is the target 1 time in 63 or so, and the walk mixes fast, so roughly 63–70 makes sense.

Interviewers may push further: The same report asked for a market on this number. How wide, given its spread?

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Quant Trading Analyst intern · Interview · Expected value · reported by a candidate

Draw 10 random chords in a circle, each joining two independent uniform points on the circumference.

Reported for k chords; we take k = 10 and chords whose endpoints are uniform on the circle.

1. What is the expected number of intersection points between them?
   Answer and worked solution

   Answer: 15

   Key idea. Linearity over pairs of chords: each pair crosses with probability 1/3.

   1. Two chords use 4 random points. Those 4 points can be paired into 2 chords in 3 ways, equally likely, and exactly one pairing crosses.
   2. There are C(10, 2) = 45 pairs.
   3. E = 45 × 1/3 = 15.

   Check. In general: C(k, 2)/3.

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Trader intern · Interview · Mental maths · reported by a candidate

Mental multiplication, asked out loud.

1. 73 × 74
   Answer and worked solution

   Answer: 5402

   Key idea. Square of the first number, plus the first number.

   1. 73 × 74 = 73² + 73.
   2. 73² = 5329 (70² + 2·70·3 + 9).
   3. 5329 + 73 = 5402.

   Check. Ends in 2 (3 × 4 = 12).

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