Study elegant solutions

Three Heads Before Three Tails

A fair coin is tossed until either three consecutive heads or three consecutive tails appear. What is the probability heads wins?

Coins

Balanced Committee

From 6 women and 5 men, how many 4-person committees contain exactly 2 women?

Committees

Precision of the Mean

A population has variance $16$ . For a random sample of $64$ independent observations, what is the standard error of the sample mean?

Estimation

Two Heads in Three Flips

A fair coin is flipped three times. What is the probability of getting exactly two heads?

Coins

At Least One Six

A fair six-sided die is rolled four times. What is the probability of seeing at least one 6?

Dice

Sum of Eight

Two fair six-sided dice are rolled. What is the probability that their sum is 8?

Dice

Matching Ranks

Two cards are drawn without replacement from a standard 52-card deck. What is the probability that they have the same rank?

Cards

Even Prime Sum

Two distinct prime numbers are chosen uniformly from the primes between 1 and 20 inclusive. What is the probability that their sum is even?

Number Theory

Two Winning Trades

Each trade independently makes money with probability 0.4. What is the probability that exactly two out of five trades make money?

Binomial

No Correct Letter

Four letters are randomly placed into four addressed envelopes, one letter per envelope. What is the probability that no letter reaches the correct recipient?

Permutations

Two Red Balls

A bag contains 3 red balls and 7 blue balls. Five balls are chosen without replacement. What is the probability of selecting exactly two red balls?

Sampling Without Replacement

Rolling Doubles

Two fair six-sided dice are rolled. What is the probability of rolling doubles?

Dice

Boxes With Minimums

15 identical balls are distributed among 5 labeled boxes. Each box must contain at least 2 balls. How many distributions are possible?

Stars and Bars

Four Boxes, Min Three

20 identical balls are distributed among 4 labeled boxes. Each box must contain at least 3 balls. How many distributions are possible?

Stars and Bars

Non-Adjacent Books

Five distinct books are placed on a shelf. Two particular books must not be adjacent. How many valid arrangements are there?

Arrangements

MISSISSIPPI Arrangements

How many distinct arrangements of the letters in MISSISSIPPI are possible?

Repeated Objects

Committee With Alice or Bob

A committee of 4 is selected from 10 people. Alice and Bob are two of the ten people. How many committees contain at least one of Alice or Bob?

Committees

Round Table Seating

+ 2 follow-ups

Eight distinct people sit around a circular table. Rotations are considered identical. How many seating arrangements are possible?

Circular Permutations

Bounded Integer Solutions

How many integer solutions satisfy $x_{1} + x_{2} + x_{3} = 20$ with $x_{1} \geq 2$ , $x_{2} \geq 3$ , $x_{3} \geq 4$ ?

Integer Solutions

Max Below a Threshold

$X_{1}$ , $X_{2}$ , and $X_{3}$ are independent $Uniform (0, 1)$ random variables. What is the probability that their maximum is less than $0.8$ ?

Maximum

Sum of Independent Normals

$X \sim N (μ_{1}, σ_{1}^{2})$ and $Y \sim N (μ_{2}, σ_{2}^{2})$ are independent. What is the distribution of $X + Y$ ?

Normal Distribution

Two Boxes of Balls

You choose one of two boxes uniformly. Box A contains 2 red balls and 1 blue ball. Box B contains 1 red ball and 2 blue balls. You draw a red ball. What is the probability that you chose Box A?

Bayes

First Ace to Appear

A standard deck is shuffled. Among the four aces, which ace is most likely to appear first?

Cards

Dice Sum Conditional

Two fair six-sided dice are rolled. Given that their sum is 8, what is the probability that at least one die shows 5?

Dice

Monty Hall Switch

In the standard three-door Monty Hall problem, you initially choose one door. The host, who knows where the prize is, opens a losing door and offers you the opportunity to switch. What is your probability of winning if you switch?

Monty Hall

Symmetric Walk Mean

A symmetric random walk starts at $0$ . Each step is independently $+ 1$ or $- 1$ with equal probability. What is the expected position after $100$ steps?

Random Walk

Waiting for Sixes

What is the expected number of rolls of a fair six-sided die until the first $6$ appears?

Waiting Times

Binary Payout Fair Value

A contract pays $$ 100$ with probability 0.6 and $$ 0$ otherwise. Assume zero interest rates and a risk-neutral setting. What is its fair value?

Markets · Fair Value

Uninformed Spread Profit

The true fair value of an asset is 100. A market maker quotes 99 bid and 101 ask. An uninformed trader is equally likely to buy at the ask or sell at the bid. Ignoring inventory risk, what is the market maker's expected profit per trade?

Market Making · Spread

Annualized Sharpe Ratio

A strategy has an average daily return divided by daily return volatility equal to 0.10. Assuming independent identically distributed daily returns and 252 trading days per year, what is the annualized Sharpe ratio?

Risk · Sharpe Ratio

Duration Price Sensitivity

A bond has modified duration 5. Interest rates increase by 20 basis points. Using the duration approximation, approximately what percentage change should occur in the bond price?

Fixed Income · Duration

Delta-Neutral Share Count

You are long 100 call option contracts. Each contract represents 100 shares and each option has delta 0.60. How many shares should you trade to become delta-neutral?

Options · Delta Hedging

No Consecutive Heads

A fair coin is flipped five times. What is the probability that no two heads occur consecutively?

Coins

Relative Order

Eight distinct people stand in a uniformly random order. What is the probability that Alice appears before Bob and Bob appears before Charlie?

Random Permutations

Equal Team Allocation

Twelve distinct people are assigned to three labeled teams, with exactly four people on each team. How many assignments are possible?

Multinomial Counting

Adjacent Equal Coin Flips

A fair coin is flipped 20 times. Let $X$ be the number of adjacent pairs of flips that are equal. For example, flips 7 and 8 form one adjacent pair. Find $E [X]$ .

Indicator Variables

Small Triangle in the Unit Square

$X$ and $Y$ are independent $Uniform (0, 1)$ random variables. What is $P (X + Y < \frac{1}{2})$ ?

Geometric Probability

Population Regression Coefficient

Suppose $Cov (X, Y) = 6$ and $Var (X) = 4$ . What is the population least-squares slope when regressing $Y$ on $X$ ?

Linear Regression

Eigenvalues of a Symmetric 2x2 Matrix

Find the eigenvalues of the matrix $(2112)$ .

Linear Algebra

Rank of an Outer Product

Let $u$ and $v$ be nonzero vectors. What is the rank of the matrix $u v^{T}$ ?

Linear Algebra

Minimum Norm Under a Linear Constraint

Minimize $x^{2} + y^{2}$ subject to $x + y = 1$ .

Optimization

Superposition of Two Poisson Processes

Two independent Poisson processes have rates $λ_{1}$ and $λ_{2}$ . What is the distribution of their combined arrival process?

Poisson Processes

Forward Price With a Dividend Yield

A stock trades at $100$ . The continuously compounded risk-free rate is $5%$ and the stock pays a continuous dividend yield of $2%$ . What is the no-arbitrage one-year forward price?

Forwards

Compounded Two-Period Return

A stock moves from $100$ to $110$ , then from $110$ to $99$ . What is the total simple return over the two periods?

Returns

Worst-Case Binary Search Comparisons

A sorted array contains 1000 distinct values. Using binary search, what is the maximum number of midpoint comparisons needed to determine whether a target value is present?

Three Heads in Four Flips

A fair coin is flipped four times. What is the probability of obtaining exactly three heads?

Coins

At Least One Six in Three Rolls

A fair six-sided die is rolled three times. What is the probability that at least one roll is a 6?

Dice

Dice Sum of Nine

Two fair six-sided dice are rolled. What is the probability that their sum is 9?

Dice

Matching Parity of Two Dice

Two fair six-sided dice are rolled. What is the probability that both results have the same parity?

Dice

Waiting More Than Five Flips

A fair coin is flipped until the first head appears. What is the probability that more than five flips are required?

Geometric Distribution

First Head on Flip Four

A coin lands heads with probability 0.3 on each flip. What is the probability that the first head occurs on flip 4?

Geometric Distribution

Divisible by Two or Five

An integer is selected uniformly at random from ${1, \dots, 100}$ . What is the probability that it is divisible by 2 or 5?

Number Theory

Relative Order of Three Objects

A uniformly random permutation of six distinct objects is generated. What is the probability that object A occurs before object B and object B occurs before object C?

Random Permutations

Which Variable Is Largest

X, Y and Z are independent, identically distributed continuous random variables. What is the probability that X is the largest of the three?

Order Statistics

Repeated Digit in a Random Code

A five-digit code is generated by independently selecting each digit uniformly from 0 through 9. What is the probability that at least one digit is repeated?

Combinatorial Probability

Exactly Two Red Balls

A bag contains 4 red and 6 blue balls. Three balls are drawn without replacement. What is the probability that exactly two are red?

Hypergeometric Distribution

Committees of Three

How many committees of 3 can be chosen from 8 distinct people?

Combinations

Arranging Seven Books

How many ways can seven distinct books be arranged on a shelf?

Permutations

Circular Table for Six

Six distinct people sit around a circular table. Rotations are considered identical. How many seatings exist?

Circular Permutations

Seven in a Line, Adjacent

Seven people stand in a line. How many arrangements have two particular people, Alice and Bob, standing adjacent?

Permutations

Arrangements of BANANA

How many distinct arrangements of the letters in the word BANANA exist?

Multiset Permutations

Balls Into Boxes, Empty Allowed

Ten identical balls are distributed among four labeled boxes, with empty boxes allowed. How many allocations exist?

Stars and Bars

Balls Into Boxes, None Empty

Ten identical balls are distributed among four labeled boxes, with every box receiving at least one ball. How many allocations exist?

Stars and Bars

Five-Digit Codes, No Repeats

How many length-5 codes can be made from digits 0 through 9 if no digit may repeat and a leading zero is allowed?

Permutations

Lattice Paths to (5,3)

How many lattice paths travel from $(0, 0)$ to $(5, 3)$ using only right and up moves?

Lattice Paths

Expected Heads in Twenty Flips

A fair coin is flipped 20 times. What is the expected number of heads?

Linearity

Expected Heads with a Biased Coin

A coin has head probability 0.3 and is flipped 50 times. What is the expected number of heads?

Linearity

Expected Sum of Ten Dice

Ten fair six-sided dice are rolled. What is the expected sum?

Linearity

Waiting for a Rare Success

A Bernoulli trial succeeds with probability 0.2. What is the expected number of trials until the first success?

Geometric Distribution

Random Walk Mean After Twenty Steps

A symmetric +/-1 random walk starts at 0. What is its expected position after 20 steps?

Random Walk

Expected Failures Before Success

Independent Bernoulli trials succeed with probability 1/4. What is the expected number of failures before the first success?

Geometric Distribution

Sum of Two Independent Normals

$X \sim N (2, 4)$ and $Y \sim N (3, 9)$ are independent. What is the distribution of $X + Y$ ?

Normal Distribution

Linear Transformation of a Normal

$X \sim N (μ, σ^{2})$ . What is the distribution of $a X + b$ ?

Normal Distribution

MLE of a Bernoulli Parameter

A Bernoulli sample contains $k$ successes in $n$ observations. What is the maximum-likelihood estimator of $p$ ?

Maximum Likelihood

MLE of a Normal Mean

$X_{1}, \dots, X_{n}$ are independent $N (μ, σ^{2})$ observations with $σ$ known. What is the maximum-likelihood estimator of $μ$ ?

Maximum Likelihood

Variance of the Sample Mean

$X_{1}, \dots, X_{n}$ are independent with variance $σ^{2}$ . What is $Var (X ˉ)$ for $n$ observations?

Sampling Distributions

Large-Sample Distribution of the Sample Mean

A population has mean $μ$ and variance $σ^{2}$ . State the approximate large- $n$ distribution of the sample mean.

Central Limit Theorem

Posterior Mean of a Beta Distribution

Using the posterior $Beta (7, 8)$ , what is the posterior mean?

Bayesian Inference

Mean and Variance of a Poisson

If $X \sim Poisson (λ)$ , state its mean and variance.

Discrete Distributions

Mean and Variance of an Exponential

If $X \sim Exponential (λ)$ , state its mean and variance.

Continuous Distributions

Mean of a Geometric Waiting Time

For a geometric random variable counting trials until the first success with probability $p$ , what is the expected value?

Discrete Distributions

Mean of a Uniform Distribution

$X \sim Uniform (a, b)$ . Find $E [X]$ .

Continuous Distributions

Variance of a Uniform Distribution

$X \sim Uniform (a, b)$ . Find $Var (X)$ .

Continuous Distributions

Marginal Distribution of Brownian Motion

For standard Brownian motion $B_{t}$ with $B_{0} = 0$ , what is the distribution of $B_{2}$ ?

Brownian Motion

Distribution of a Brownian Increment

For standard Brownian motion, what is the distribution of $B_{5} - B_{2}$ ?

Brownian Motion

Covariance of Two Brownian Times

For standard Brownian motion, find $Cov (B_{2}, B_{7})$ .

Brownian Motion

Symmetry of Brownian Motion

For standard Brownian motion, what is $P (B_{1} > 0)$ ?

Brownian Motion

Second Moment of Brownian Motion

For standard Brownian motion, find $E [B_{t}^{2}]$ .

Brownian Motion

Poisson Process Counts

A counting process $N (t)$ is a Poisson process with rate $λ$ . What is the distribution of $N (t)$ ?

Poisson Process

Interarrival Times of a Poisson Process

For a Poisson process with rate $λ$ , what is the distribution of the time between consecutive arrivals?

Poisson Process

Superposition of Two Rate-Specific Poisson Processes

Two independent Poisson processes have rates $4$ and $7$ arrivals per hour. What is the rate of their superposition (the combined arrival process)?

Poisson Process

Expected Time to the nth Poisson Arrival

For a Poisson process with rate $λ$ , what is the expected waiting time until the $n$ th arrival?

Poisson Process

Forward Price With Cost of Carry

A stock trades at $100$ . The continuously compounded risk-free rate is $4%$ , the continuous dividend yield is $1%$ , and maturity is one year. Find the no-arbitrage forward price.

Forwards

Duration-Only Price Sensitivity, Second Instance

A bond has modified duration 6. Yields rise 25 basis points. Approximate the percentage price change using duration only.

Fixed Income · Duration

Continuously Compounded Zero-Coupon Bond Price

A zero-coupon bond pays 100 in two years. The continuously compounded risk-free rate is 5%. Find its price.

Fixed Income

Expected P&L of a Two-Outcome Strategy

A strategy earns +4 with probability 0.4 and loses 2 otherwise. Find the expected P&L per trade.

Trading

Uninformed Spread Profit, Second Instance

True fair value is 100. A market maker quotes 99.80/100.20. An uninformed customer is equally likely to buy or sell. Ignoring inventory risk, what is expected maker profit per trade?

Market Making · Spread

One-Day Value at Risk, Second Instance

Daily P&L is normally distributed with zero mean and $1.5 million standard deviation. Using $z = 2.33$ , estimate the one-day 99% VaR.

Risk · Value at Risk

Delta-Neutral Share Count, Second Instance

A call has delta 0.35. You are long 200 contracts, each representing 100 shares. How many shares should be traded to become delta-neutral?

Options · Delta Hedging

Eigenvalues of a Symmetric 2x2 Matrix

Find the eigenvalues of the matrix $(3113)$ .

Eigenvalues

Trace From Eigenvalues

A matrix has eigenvalues $2$ , $4$ , and $5$ . What is its trace?

Eigenvalues

Determinant From Eigenvalues

A matrix has eigenvalues $2$ , $4$ , and $5$ . What is its determinant?

Eigenvalues

Condition for Orthogonal Vectors

What condition characterizes orthogonality of two vectors $x$ and $y$ ?

Inner Products

Minimizing Sum of Squares Under a Linear Constraint

Minimize $x^{2} + y^{2}$ subject to $x + y = 2$ .

Optimization

Maximizing a Product Under a Sum Constraint

Maximize $x y$ subject to $x + y = 10$ and $x, y \geq 0$ .

Optimization

Normal Equations for Ordinary Least Squares

State the normal equations for ordinary least squares.

Least Squares

Eigenvalues of a Triangular Matrix

What are the eigenvalues of an upper triangular matrix?

Eigenvalues

Determinant of a Triangular Matrix

What is the determinant of a triangular matrix?

Determinants

Rank-Nullity for a Map From R^7

A linear map from $R^{7}$ has rank $4$ . What is its nullity?

Rank-Nullity Theorem

Inverse of an Orthogonal Matrix

If $Q$ is an orthogonal matrix, what is $Q^{- 1}$ ?

Orthogonal Matrices

Binary Search Comparisons for 1024 Elements

A target is known to be present in a sorted array of 1024 distinct elements. What is the minimum worst-case number of yes/no comparisons required to identify its position?

Time Complexity of Merge Sort

What is the asymptotic time complexity of merge sort?

Sorting

Two-Sum With a Hash Set

What is the time complexity of finding whether a target pair summing to a given value exists using a hash set?

Hashing

Solving T(n) = T(n/2) + 1

Solve $T (n) = T (n /2) + 1$ .

Recurrences

Solving T(n) = T(n-1) + 1

Solve $T (n) = T (n - 1) + 1$ .

Recurrences

Maximum Nodes in a Binary Tree of Bounded Depth

A binary tree has root depth 0 and maximum depth d. What is the maximum possible number of nodes?

Trees

Binary Heap Insertion Complexity

What is the worst-case time complexity of inserting an element into a binary heap?

Heaps

Shortest Paths in an Unweighted Graph

What algorithm finds shortest paths from a source in an unweighted graph?

Graph Traversal

Edge Weight Requirement for Dijkstra's Algorithm

What restriction on edge weights is required by standard Dijkstra's algorithm?

Shortest Paths

Existence of a Topological Ordering

When does a directed graph admit a topological ordering?

Graph Theory

Guaranteed Repeats by the Pigeonhole Principle

An array of n+1 integers contains values only from {1,...,n}. What principle guarantees at least one repeated value?

Combinatorial Reasoning

Expected Hash Table Lookup Complexity

What is the asymptotic expected lookup complexity of a well-designed hash table under standard assumptions?

Hashing

Maximum Edges in a Simple Graph

What is the worst-case number of edges in a simple undirected graph with n vertices?

Graph Theory

Identifying a Tree by Edge Count

A connected undirected graph contains n vertices and exactly n-1 edges. What structure must it be?

Graph Theory

Midpoint and Spread of a Quote

A dealer quotes $99.80$ bid and $100.20$ ask. What are the midpoint and the full spread?

Quotes and Spreads

Recovering the Ask

The bid is $49.70$ and the full spread is $0.60$ . What is the ask?

Quotes and Spreads

Round Trip Through the Spread

You buy $200$ shares at the bid of $19.90$ and immediately resell them at the ask of $20.10$ . Ignoring costs, what is your P&L?

P&L

Inventory After a Customer Buy

A customer buys $80$ units from you at your $12.10/12.30$ quote. What is the change in your inventory?

Inventory

Covering Part of a Short

You are short $150$ shares. A customer sells you $70$ shares. What is your new position?

Inventory

Spread in Basis Points

A dealer posts $100.00/100.50$ . Express the full spread in basis points of the midpoint.

Quotes and Spreads

Two Customers, Both Sides

Your fair value is $50$ and your quote is $49.80/50.20$ . One customer buys $10$ units and another sells $10$ units. What is your combined gross P&L relative to fair value?

P&L

Picked Off on a Stale Quote

Fair value rises from $80$ to $81$ while your quote stays at $79.80/80.20$ . A customer buys $100$ units at your stale ask. Marking to the new fair value, what is your P&L?

Adverse Selection

Realized P&L on a Round Trip

A dealer buys $40$ units at $8.20$ , then sells them at $8.45$ . Find the realized P&L.

P&L

Half-Spread in Basis Points

At bid/ask $75/77$ , what is the half-spread, in price units and in basis points of the midpoint?

Quotes and Spreads

Three Switches, One Visit

Three switches outside a closed room control three incandescent bulbs inside. You may enter the room exactly once. How do you identify which switch controls each bulb?

Logic Puzzles

Eight Balls, One Heavy

Eight identical-looking balls include one heavier ball. Using a balance scale, what is the minimum number of weighings that guarantees finding the heavy ball?

Weighing

Burning Ropes for 45 Minutes

Two ropes each burn completely in exactly $60$ minutes, but at irregular rates. How can you measure exactly $45$ minutes?

Timing Puzzles

Four Liters From 3 and 5

You have an unmarked $3$ -liter jug and an unmarked $5$ -liter jug, unlimited water, and may empty either jug. Measure exactly $4$ liters.

Jug Puzzles

Snail in a Well

A snail climbs $3$ meters each day and slides down $2$ meters each night. On which day does it leave a $10$ -meter well?

Classic Puzzles

Clock Angle at 3:15

At $3 : 15$ , what is the smaller angle between the hands of an analog clock, in degrees?

Clocks

Machines and Widgets

Five identical machines make five widgets in five minutes. How long do $100$ machines need to make $100$ widgets, in minutes?

Rates

Bat and Ball

A bat and a ball cost $1.10 together. The bat costs $1 more than the ball. How much does the ball cost, in dollars?

Classic Puzzles

Three Sandwiches, Two Fathers, Two Sons

Two fathers and two sons share exactly three sandwiches so that each gets one whole sandwich. How is this possible?

Logic Puzzles

Doubling Lily Pads

A lily patch doubles its area each day and covers the entire pond on day $30$ . On which day was the pond half covered?

Exponential Growth

Even Roll

A fair six-sided die is rolled. What is the probability of an even result?

Dice

Three Heads in Four Flips

A fair coin is flipped four times. What is the probability of exactly three heads?

Coins

Two Dice Summing to Seven

Two fair dice are rolled. What is the probability that the sum is $7$ ?

Dice

Two Aces

Two cards are drawn without replacement from a standard $52$ -card deck. What is the probability both are aces?

Cards

Choosing Three of Ten

How many ways are there to choose $3$ people from $10$ ?

Combinations

Books on a Shelf

How many ways can $5$ distinct books be arranged on a shelf?

Permutations

Mean of Four Numbers

Find the mean of $2, 4, 6, 8$ .

Descriptive Statistics

Variance of a Coin Indicator

A fair coin is coded $1$ for heads, $0$ for tails. Find the variance.

Bernoulli

19 Times 21

Compute $19 \times 21$ mentally.

Mental Math

Fifteen Percent of 240

Compute $15%$ of $240$ .

Mental Math

48 Squared

Compute $4 8^{2}$ .

Mental Math

One Eighth as a Percent

Express $1/8$ as a percentage.

Mental Math

Long Call Profit

A long European call has strike $100$ and premium $5$ . The underlying expires at $112$ . What is the net profit per share?

Option Payoffs

Long Put Profit

A long European put has strike $80$ and premium $3$ . The underlying expires at $70$ . What is the net profit per share?

Option Payoffs