Desk overview

Welcome.

Small sessions, every day. That's how quant interviews are won.

Training desks

Recent sessions — accuracy

Achievements

Desk 01 · Speed

Mental Math

Arithmetic under the clock — the entry ticket to every trading interview.

Desk 02 · Probability, EV & more

The questions that actually get asked — Bayes, expected value, combinatorics, ruin, strategy — with full worked solutions.

Desk 03 · Think slow

Wrestle before you peek. Hints cost a little edge; the reveal costs all of it.

Desk 04 · Orders of magnitude

Decompose, estimate, sanity-check. Within 3× of reference is a win.

Desk 05 · Flow

Quote a two-sided market on a hidden quantity. Earn the spread, dodge the sharks, respect your inventory.

Desk 06 · The real thing

Twenty minutes, three timed sections, one verdict. Sit it like it counts.

Review

Where your edge comes from — and where it leaks. Weakest categories float to the top.

Review

Every desk on one tape. Edge is earned, not given — climb.

Reference

Results worth knowing cold. Skim before an interview; internalize over time.

Probability toolkit

Law of total probability — P(A) = Σ P(A|Bᵢ)·P(Bᵢ). Condition on the first step; most "expected time" puzzles fall to this.
Bayes — P(H|E) = P(E|H)·P(H) / P(E). Posterior odds = prior odds × likelihood ratio. Never forget the prior.
Linearity of expectation — E[X+Y] = E[X]+E[Y], no independence needed. Break nasty quantities into indicator variables.
Indicators — E[1_A] = P(A). "Expected number of …" = sum of probabilities.
Tail-sum — for non-negative integer X: E[X] = Σ P(X ≥ k). Kills E[max] and waiting-time questions.
Variance rules — Var(aX+b) = a²Var(X); Var(X+Y) = Var X + Var Y + 2Cov(X,Y).
Symmetry & exchangeability — "position k is an ace" = 1/13 for every k; "last ball is red" = fraction of reds. Look for it before computing.
Memorylessness — geometric and exponential only. The past wait is sunk.
Optional stopping — for a martingale with a bounded stopping time, E[X_T] = E[X₀]. Instant gambler's-ruin probabilities.

Numbers to know cold

Quantity	Value
E[flips to first H] / to HH / to HT	2 · 6 · 4
E[rolls to first 6]	6
E[rolls to see all 6 faces] (coupon)	14.7 = 6·H₆
Birthday: P(match) > ½	23 people
E[max of 2 dice] / E[sum of 2 dice]	161/36 ≈ 4.47 · 7
Var of one die (σ)	35/12 ≈ 2.92 (σ ≈ 1.71)
P(no fixed point), n large (derangement)	1/e ≈ 36.8%
Secretary success / cutoff	1/e · skip n/e
Gambler's ruin, fair: P(reach N from i)	i / N
E[max of n Uniform(0,1)]	n/(n+1)
E\|X−Y\|, X,Y ~ U(0,1)	1/3
Broken stick forms a triangle	1/4
Meeting problem (15 min in an hour)	7/16
Ballot: A always strictly ahead	(a−b)/(a+b)
100 prisoners (cycle strategy)	1 − ln 2 ≈ 31%
Heads in 100 flips: mean ± σ	50 ± 5
Kelly, even odds	f* = p − q

Combinatorics identities

Permutations / combinations — n!/(n−k)! ordered; C(n,k) = n!/(k!(n−k)!) unordered.
Multiset permutations — n! / (n₁!·n₂!·…) — MISSISSIPPI = 11!/(4!4!2!) = 34,650.
Stars & bars — non-negative solutions of x₁+…+x_k = n: C(n+k−1, k−1).
Inclusion–exclusion — |A∪B∪C| = Σ|A| − Σ|A∩B| + |A∩B∩C|; derangements Dₙ = n!·Σ(−1)ᵏ/k!.
Catalan — Cₙ = C(2n,n)/(n+1): balanced brackets, monotone paths under the diagonal, binary trees. 1, 1, 2, 5, 14, 42…
Pigeonhole — n+1 items in n boxes force a repeat. Guarantee ⇒ think worst case.
Double counting / symmetry — count one thing two ways; or argue every position is exchangeable.

Mental math tricks

Flip percentages — a% of b = b% of a. 8% of 25 = 25% of 8 = 2.
×5, ×25, ×50 — ×10/2, ×100/4, ×100/2. And ÷5 = ×2/10.
×11 — 43 × 11 → 4_(4+3)_3 = 473 (carry when needed).
Squares near 50 — (50±d)² = 2500 ± 100d + d². So 47² = 2500 − 300 + 9 = 2209.
Squares near 100 — (100±d)² = 10000 ± 200d + d².
Difference of squares — a·b = ((a+b)/2)² − ((a−b)/2)². 18×22 = 20² − 2² = 396.
Ending in 5 — (10a+5)² = a(a+1) then append 25. 85² = 72_25.
Decimals — count total decimal places: 1.2 × 0.04 → 12×4 = 48 → 0.048.
Fractions to know — 1/8 = .125, 1/6 ≈ .1667, 1/7 ≈ .1429, 1/9 = .111…, 1/12 ≈ .0833, 1/16 = .0625.
Rule of 72 — years to double ≈ 72 / rate%. Handy for growth sanity checks.
√2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236, e ≈ 2.718, ln 2 ≈ 0.693, π ≈ 3.1416.

Market-making principles

Quote around fair value — your mid should track conditional expected value given everything revealed so far.
Spread prices two risks — variance of the underlying and the chance your counterparty knows more (adverse selection). More of either ⇒ wider.
Getting filled is information — an instant, full-size lift means fair value is probably above your ask (winner's curse). Update, then re-quote.
Skew for inventory — long ⇒ shift both quotes down to sell off; short ⇒ shift up. Manage risk with the mid, not only the width.
Too wide never trades — no flow, no edge. Market making is selling liquidity; price it, don't hide it.
Size like Kelly — edge/odds bounds your size; half-Kelly is the practitioner's comfort zone.
Say your market out loud — in interviews: "I'm 17 at 20, five up" beats a silent number. Communicate width, level, and size.

Distribution cheat table

Distribution	Mean	Variance	Use it for
Bernoulli(p)	p	p(1−p)	indicators
Binomial(n,p)	np	np(1−p)	#successes
Geometric(p)	1/p	(1−p)/p²	waiting for first success
Poisson(λ)	λ	λ	rare-event counts
Uniform(0,1)	1/2	1/12	geometric probability
Exponential(λ)	1/λ	1/λ²	memoryless waits
Normal(μ,σ²)	μ	σ²	CLT; 68 / 95 / 99.7

Interview game plan

Restate the problem in one sentence and confirm the setup — half the failures are answering a different question.
Announce a plan before diving: "I'll condition on the first flip" earns points even before any algebra.
Think aloud, cleanly — interviewers grade the path. Silence reads as being lost; narrated dead-ends read as process.
Try small cases — n = 1, 2, 3 reveals the pattern in most puzzles.
Sanity-check — units, bounds (probabilities in [0,1]), limits (p→0, n→∞), symmetry.
When stuck — say what you know, what you'd try, and what feels wrong. Ask for a hint after real effort; using a hint well is a strong signal.
Behavioral prep — have crisp stories for: a failure, a disagreement, a decision under uncertainty, why trading/research, why this firm. 90 seconds each.
Own your resume — every line can be dissected. If you can't derive it, cut it.