Problems

118 curated questions across 15 topics, each with hints and a worked solution, including at 24 firms.

– of 118 solved

Easy –/53Medium –/52Hard –/13

/

80 results

Probability & Combinatorics

Medium#1
Waiting for Two Heads in a Row
Goldman Sachs · Morgan Stanley · Two Sigma
Probability · Numerical · 8 min
Easy#4
At Least One Boy
D. E. Shaw
Probability · Multiple choice · 3 min
Medium#48
Which Coin Did I Draw?
Jane Street
Probability · Numerical · 6 min
Medium#49
HHT or HTT First?
Jane Street
Probability · Numerical · 8 min
Easy#50
An Even Number of Heads
Jane Street
Probability · Numerical · 3 min
Medium#55
A Fair Flip from a Biased Coin
Jane Street · Akuna Capital
Probability · Numerical · 6 min
Easy#58
P(X > Y) for Two Normals
Citadel
Probability · Numerical · 3 min
Medium#59
Waiting for Two Sixes in a Row
Citadel Securities
Probability · Numerical · 6 min
Medium#61
Best of Three or Win by Two?
Susquehanna (SIG) · Optiver
Probability · Multiple choice · 6 min
Easy#65
Will the Next Card Be Higher?
Susquehanna (SIG)
Probability · Numerical · 4 min
Easy#66
Three Dice in Increasing Order
Optiver
Probability · Numerical · 3 min
Hard#68
Break the Longer Piece
Goldman Sachs
Probability · Numerical · 12 min
Easy#71
First to Roll a Six
J.P. Morgan
Probability · Numerical · 4 min
Easy#74
Simulate a Die with a Coin
J.P. Morgan · Jane Street
Probability · Free response · 6 min
Easy#78
A Sum of 4 Before a Sum of 2
Goldman Sachs
Probability · Numerical · 3 min
Medium#79
Product of Two Uniforms Below One Half
Goldman Sachs
Probability · Numerical · 6 min
Medium#82
Every Even Face Before Any Odd
IMC Trading
Probability · Numerical · 5 min
Easy#83
Four-Sided or Six-Sided?
IMC Trading
Probability · Numerical · 4 min
Easy#85
A Seven Somewhere Up to a Million
Optiver
Probability · Numerical · 3 min
Easy#88
Meeting at the Station
Citadel
Probability · Numerical · 4 min
Easy#94
One Double-Headed Coin in a Thousand
Goldman Sachs · Squarepoint Capital · Renaissance Technologies
Probability · Numerical · 4 min
Easy#96
One Third from a Fair Coin
Goldman Sachs
Probability · Free response · 5 min
Easy#100
One Fair Coin Among Many
Jane Street · Millennium
Probability · Multiple choice · 4 min
Medium#103
51 Coins Against 50
Squarepoint Capital
Probability · Numerical · 4 min
Easy#104
Will Two Teams Meet?
Five Rings
Probability · Numerical · 4 min
Medium#108
One of Them Is Named William
D. E. Shaw
Probability · Multiple choice · 7 min
Easy#109
A Negative Test with No Base Rate
D. E. Shaw
Probability · Free response · 4 min
Easy#113
Three Flips, at Least One Head
Akuna Capital
Probability · Numerical · 3 min
Easy#115
Waiting for HHT
Jane Street · Deutsche Bank
Probability · Numerical · 5 min
Easy#118
Real Roots from Random Coefficients
Bank of America
Probability · Numerical · 4 min

Statistics & Statistical Inference

Easy#76
Uncorrelated but Not Independent
Two Sigma
Statistics · Free response · 5 min
Medium#87
Range of the Third Correlation
Citadel
Statistics · Multiple choice · 6 min
Medium#89
Best Guess from a Noisy Signal
Citadel · Susquehanna (SIG)
Statistics · Numerical · 6 min
Easy#90
Mean of a Lognormal
Citadel
Statistics · Numerical · 3 min
Hard#91
From Correlation to Win Rate
Citadel
Statistics · Numerical · 10 min
Medium#92
Duplicating Your Regression Data
Point72 · J.P. Morgan
Statistics · Multiple choice · 5 min
Easy#95
At Least 220 Heads in 400 Flips
Goldman Sachs · Squarepoint Capital
Statistics · Numerical · 4 min
Easy#101
Best-Fit Line Through Three Points
Squarepoint Capital
Statistics · Numerical · 4 min
Medium#106
Seven Dice Between 17 and 27
Five Rings
Statistics · Numerical · 4 min
Easy#117
Correlated Predictors in a Regression
Two Sigma
Statistics · Free response · 5 min

Expected Value & Decision Theory

Medium#10
Roll Once, Maybe Reroll
Citadel · Jane Street · Squarepoint Capital
Expected Value · Numerical · 6 min
Medium#47
Die Game with Two Rerolls
Jane Street · Susquehanna (SIG)
Expected Value · Numerical · 6 min
Easy#51
The Advantage of Flipping First
Jane Street
Expected Value · Numerical · 4 min
Medium#52
Thirty-Sided Die vs Twenty-Sided Die
Jane Street
Expected Value · Numerical · 8 min
Medium#53
Square, Product or Median Squared?
Jane Street
Expected Value · Multiple choice · 6 min
Hard#54
100-Sided Die, Rerolls Cost a Dollar
Jane Street
Expected Value · Numerical · 12 min
Hard#56
Add One or Invert
Jane Street
Expected Value · Numerical · 15 min
Hard#60
Seeing Every Face Twice
Citadel Securities
Expected Value · Numerical · 15 min
Easy#63
Accepting a Doubled Bet
Susquehanna (SIG)
Expected Value · Numerical · 4 min
Easy#64
Even Faces Pay Double
Susquehanna (SIG)
Expected Value · Numerical · 5 min
Hard#67
Tying Spaghetti into Loops
Hudson River Trading
Expected Value · Numerical · 10 min
Easy#72
Counting Switches in Coin Flips
J.P. Morgan
Expected Value · Numerical · 3 min
Easy#75
Twenty per Head or Five Hundred for Four
Akuna Capital
Expected Value · Multiple choice · 3 min
Medium#77
Uniforms Until the Sum Exceeds One
Two Sigma
Expected Value · Numerical · 8 min
Easy#81
Paid for Every Point Over Ten
DRW
Expected Value · Numerical · 5 min
Medium#84
Three Consecutive Hidden Cards
Optiver
Expected Value · Numerical · 8 min
Easy#98
Expected Gap Between Two Dice
Jane Street
Expected Value · Numerical · 3 min
Easy#102
Average Maximum of Two Uniforms
Squarepoint Capital
Expected Value · Numerical · 3 min
Medium#110
Apples Left in the Basket
WorldQuant
Expected Value · Numerical · 5 min

Brain Teasers & Logic

Medium#57
Guess 100 Flips with One Question
Jane Street
Brain Teasers · Multiple choice · 5 min
Medium#93
A Hundred Ants on a Stick
J.P. Morgan
Brain Teasers · Numerical · 7 min
Medium#97
Fastest Three of 25 Horses
Goldman Sachs
Brain Teasers · Numerical · 8 min
Easy#105
Most Pizza Slices from Ten Cuts
Five Rings
Brain Teasers · Numerical · 3 min
Easy#112
Right Angles on a Clock
WorldQuant
Brain Teasers · Numerical · 3 min

Mental Math & Estimation

Easy#99
Digits in 99 to the 20th
Jane Street
Mental Math · Numerical · 2 min

Market Making & Trading Games

Medium#21
Fair Value of the Higher Die
Jane Street
Market Making · Numerical · 5 min
Medium#114
A Guaranteed Bet on Three Horses
Susquehanna (SIG)
Market Making · Numerical · 5 min

Stochastic Processes & Markov Chains

Easy#62
One Dollar Against Two
Susquehanna (SIG)
Stochastic Processes · Numerical · 5 min
Medium#80
Paint the Second Ball
Jump Trading
Stochastic Processes · Numerical · 8 min
Hard#86
Will the Amoebas Die Out?
Morgan Stanley
Stochastic Processes · Numerical · 10 min
Easy#107
Up Ten Before Down Five
Five Rings
Stochastic Processes · Numerical · 3 min

Stochastic Calculus

Medium#26
Expected Exponential of Brownian Motion
J.P. Morgan
Stochastic Calculus · Numerical · 5 min
Medium#27
Show W² − t Is a Martingale
J.P. Morgan
Stochastic Calculus · Free response · 10 min
Medium#69
Hit +1 Before −2
Morgan Stanley · Credit Suisse
Stochastic Calculus · Numerical · 6 min
Medium#70
Which Powers of W Are Martingales?
Morgan Stanley
Stochastic Calculus · Multiple choice · 5 min
Medium#73
Making W³ a Martingale
J.P. Morgan
Stochastic Calculus · Multiple choice · 6 min

Derivatives Pricing & Greeks

Medium#116
Interest Rates and Early Exercise
Two Sigma
Derivatives · Free response · 6 min

Linear Algebra & Optimization

Medium#35
Two-Asset Minimum-Variance Portfolio
Morgan Stanley
Linear Algebra · Numerical · 6 min
Medium#36
How Negative Can Three Correlations Be?
Jump Trading · Citadel Securities
Linear Algebra · Numerical · 8 min
Easy#111
Determinant of a Random Symmetric Matrix
WorldQuant
Linear Algebra · Numerical · 4 min