Problems

Waiting for Two Heads in a Row

Goldman Sachs · Morgan Stanley · Two Sigma

Expected Flips Until HTH

Birthday Problem Threshold

At Least One Boy

D. E. Shaw

Monty Hall with Four Doors

Positive Test, Rare Condition

Which Coin Did I Draw?

Jane Street

HHT or HTT First?

Jane Street

An Even Number of Heads

Jane Street

A Fair Flip from a Biased Coin

Jane Street · Akuna Capital

P(X > Y) for Two Normals

Citadel

Waiting for Two Sixes in a Row

Citadel Securities

Break the Longer Piece

Goldman Sachs

First to Roll a Six

J.P. Morgan

Simulate a Die with a Coin

J.P. Morgan · Jane Street

A Sum of 4 Before a Sum of 2

Goldman Sachs

Product of Two Uniforms Below One Half

Goldman Sachs

Every Even Face Before Any Odd

IMC Trading

Four-Sided or Six-Sided?

IMC Trading

A Seven Somewhere Up to a Million

Optiver

Meeting at the Station

Citadel

One Double-Headed Coin in a Thousand

Goldman Sachs · Squarepoint Capital · Renaissance Technologies

One Third from a Fair Coin

Goldman Sachs

One Fair Coin Among Many

Jane Street · Millennium

51 Coins Against 50

Squarepoint Capital

Will Two Teams Meet?

Five Rings

One of Them Is Named William

D. E. Shaw

A Negative Test with No Base Rate

D. E. Shaw

Waiting for HHT

Jane Street · Deutsche Bank

Real Roots from Random Coefficients

Bank of America

Is the Coin Fair?

Why Divide by n − 1?

Regressing Both Ways

Uncorrelated but Not Independent

Two Sigma

Range of the Third Correlation

Citadel

Best Guess from a Noisy Signal

Citadel · Susquehanna (SIG)

Mean of a Lognormal

Citadel

From Correlation to Win Rate

Citadel

Duplicating Your Regression Data

Point72 · J.P. Morgan

At Least 220 Heads in 400 Flips

Goldman Sachs · Squarepoint Capital

Best-Fit Line Through Three Points

Squarepoint Capital

Seven Dice Between 17 and 27

Five Rings

Correlated Predictors in a Regression

Two Sigma

Roll Once, Maybe Reroll

Citadel · Jane Street · Squarepoint Capital

Hats Returned at Random

Seeing Every Face

Die Game with Two Rerolls

Jane Street · Susquehanna (SIG)

Thirty-Sided Die vs Twenty-Sided Die

Jane Street

Square, Product or Median Squared?

Jane Street

100-Sided Die, Rerolls Cost a Dollar

Jane Street

Add One or Invert

Jane Street

Seeing Every Face Twice

Citadel Securities

Tying Spaghetti into Loops

Hudson River Trading

Counting Switches in Coin Flips

J.P. Morgan

Uniforms Until the Sum Exceeds One

Two Sigma

Three Consecutive Hidden Cards

Optiver

Expected Gap Between Two Dice

Jane Street

Average Maximum of Two Uniforms

Squarepoint Capital

Apples Left in the Basket

WorldQuant

Five Pirates Split the Gold

Guess 100 Flips with One Question

Jane Street

A Hundred Ants on a Stick

J.P. Morgan

Fastest Three of 25 Horses

Goldman Sachs

Estimate √50

Fair Value of the Higher Die

Jane Street

Why Widen Your Spread?

Gambler's Ruin with an Edge

Regime-Switching Stationary Distribution

One Dollar Against Two

Susquehanna (SIG)

Paint the Second Ball

Jump Trading

Will the Amoebas Die Out?

Morgan Stanley

Up Ten Before Down Five

Five Rings

Differential of W²

Expected Exponential of Brownian Motion

J.P. Morgan

Show W² − t Is a Martingale

J.P. Morgan

Hit +1 Before −2

Morgan Stanley · Credit Suisse

Which Powers of W Are Martingales?

Morgan Stanley

Making W³ a Martingale

J.P. Morgan

Put–Call Parity

Pricing an At-the-Money Call

Implied Volatility by Bisection

Interest Rates and Early Exercise

Two Sigma

Price a Two-Year Coupon Bond

The One-Year Rate, One Year Forward

Two-Asset Minimum-Variance Portfolio

Morgan Stanley

How Negative Can Three Correlations Be?

Jump Trading · Citadel Securities

Determinant of a Random Symmetric Matrix

WorldQuant

Stationary Variance of an AR(1)

Spot the Look-Ahead Bias

Maximum Drawdown

EWMA Volatility Filter

Exact Dice-Sum Probabilities

Standard Error of a Monte Carlo π

Monte Carlo European Call

When Do Antithetic Variates Help?