The Longest Broken Piece
Snap a stick at two random points into three pieces. On average, how much of the stick does the longest piece claim, and what is left for the middle and shortest?
Problems
Questions
Problems I work through, each with a full worked solution.
Probability & Statistics49
Four Coins Against Five
I toss four coins and you toss five. You win only by getting strictly more heads than I do. That extra coin sounds like a small edge, so how likely are you to win?
Correlation Under Shifts and Scales
The correlation of X and Y is rho. Now shift X by a constant, or multiply it by a positive number. Which of these moves the correlation, and which leaves it alone?
A Positive Test
A disease hits 0.5 percent of people, and a test never misses a real case but flags 7 percent of healthy people. A random stranger tests positive. How worried should they be?
HTH Before HHT
Flip a fair coin until either HTH or HHT shows up as three in a row. The two triplets look interchangeable, so are they really equally likely to arrive first?
The Mean of e to the X
If X is normal with mean 0 and variance sigma squared, what is the expected value of e to the X? The answer is not 1, and the gap reveals something about averaging an exponential.
Margin of Error
A poll of 1000 people puts Candidate A at 60 percent. How do you attach a margin of error to that single number?
The Normal Density
A normal variable with mean mu and variance sigma squared has the familiar bell-shaped density. What is its exact formula, and where does the constant factor in front of the exponential come from?
Pebble in Four Boxes
A pebble starts in Box 1 and a fair coin shuffles it among four boxes, ending only when it reaches Box 4. How many tosses should you expect to wait?
Standard Deviation of One Through Five
What is the standard deviation of the five numbers 1, 2, 3, 4, 5? The answer depends on one small modelling choice worth naming.
The Two-Headed Penny
A jar of 999 fair pennies hides one two-headed coin. You draw one, flip ten heads in a row, and wonder which coin you are holding. How sure can you be?
Two Kings
Deal yourself two cards from a shuffled 52-card deck that holds four kings. How likely is it that both of your cards are kings?
Product of Two Uniforms
Draw two independent numbers uniformly from the unit interval and multiply them. How likely is the product to land above one half?
Brownian Exit Time
A Brownian motion starts at 0 and runs until it first reaches +1 or -1. What is its expected exit time?
Brownian Motion and Its Square
Let B_t be a standard Brownian motion. What is the correlation between B_t and its square?
Will the Drifted Path Reach Minus One
X is a generalized Wiener process with unit upward drift, dX = dt + dW. How likely is it ever to fall as low as -1?
Cereal Toys
A cereal company hides one of four toys in each box, each equally likely. How many boxes on average do you buy to collect all four?
Positive Then Negative
Let B_t be a Brownian motion. How likely is it that B_1 is positive while B_2 is negative?
Waiting for HTH
Flip a fair coin until the pattern HTH first appears. How many flips should you expect to make?
Properties of Brownian Motion
Brownian motion underpins continuous-time finance. What conditions define it, and what striking properties follow?
Product Over One Half
Two independent uniform draws on the interval from 0 to 1. How likely is their product to exceed one half?
Sampling the d-Ball
To sample a point in the unit d-ball, draw from the surrounding cube and keep it if it lands inside. How often does that succeed as the dimension grows?
Penney's Game
Player 1 names a coin triplet, then player 2 names a different one, and you flip until one appears. Should you go first or second, and how often do you win?
Waiting for THH
Flip a fair coin until the pattern THH first appears. How many flips should you expect to make?
HHH Before THH
Flip a fair coin until either HHH or THH first appears in the sequence. How often does HHH come first?
Heads in a Row
Flip a fair coin until you see n heads in a row. How many flips should you expect to need?
The Drunk on the Bridge
A drunk stands at the 17th meter of a 100-meter bridge, staggering one meter each way with equal chance. Does he reach the far end before falling off the start, and how long does he wander first?
A Dice Duel
Roll two dice over and over. A wins on the first sum of 12, B wins on the first back-to-back pair of 7s. Who is more likely to win?
Sum of Uniforms Below One
Add up n independent uniform draws from the interval from 0 to 1. How likely is the total to land below 1?
Coupon Coverage
Each cereal box holds one of N equally likely coupons. After opening n boxes, how many distinct types does the collector expect to hold?
Coupon Collection
Each cereal box holds one of N equally likely coupons. How many boxes does it take on average to collect at least one of every type?
The First Ace
Flip a shuffled deck one card at a time. On average, how many cards turn over before the first ace appears?
Moments of the Normal
Take a standard normal random variable. What are its first four moments, and what rule connects each even one to the last?
Bus Waiting Time
Buses come on average every ten minutes, and you show up knowing nothing about the schedule. How long should you expect to wait, and how long since the last bus left?
Probability of a Triangle
A stick is snapped at two random points into three pieces. How often can those pieces be bent into a triangle?
Cars on a Road
Seeing at least one car over 20 minutes has probability 609/625. How likely is a car to pass in a 5-minute glance?
Basketball Scores
A player makes her first throw, misses her second, then scores at the rate she has scored so far. After 100 throws, how likely is she to have made exactly 50?
The Meeting Problem
Two friends, a one-hour window, a five-minute wait. How likely are they to actually meet?
Correlation of Max and Min
Two uniform draws, their minimum and their maximum. How tightly do the two ends move together?
Expected Value of Max and Min
For n independent uniform draws, where do the largest and smallest tend to fall, and what laws describe them?
Gambler's Ruin
M has a dollar, N has two, and M is the better player. Does skill at the table outrun the smaller bankroll?
Aces
Deal 52 cards evenly to four players. How often does each of the four hands hold exactly one ace?
Candies in a Jar
Ten red, twenty blue, thirty green candies drawn in random order. When the last red appears, how likely is a blue and a green to still remain?
Amoeba Population
An amoeba dies, stays, doubles, or triples each minute with equal odds. Does its line survive forever or vanish?
Dice Order
Three dice rolled one by one. How likely are the three faces to come up strictly increasing?
Fair Odds from an Unfair Coin
A coin is biased by an unknown amount. Can you still extract a perfectly fair flip from it?
Unfair Coin
One coin in a thousand is two-headed. After ten heads in a row, how sure can you be that you hold the fake?
Cubic of an Integer
A random integer up to a trillion. How often does its cube end in the digits 11?
The Monty Hall Problem
Three doors, one car. The host reveals a goat and offers a switch. Should you take it, and how often does switching win?
Brainteasers & Puzzles27
Burning Ropes
Two ropes each burn for exactly one hour, but at wildly uneven rates along their length, so half a rope need not take half an hour. Using only these ropes and a lighter, how do you time exactly 45 minutes?
The Defective Ball
Twelve identical balls hide one of a different weight, but you are not told whether it is heavier or lighter. With only a balance scale and three weighings, can you always name the odd ball and the direction it differs?
The Four Elements
Four face-down cards hold water, earth, wind, and fire. Flip them one by one, winning if you reveal both water and earth, losing the moment fire appears. How often do you win?
The Girl at the Door
I have two children and a girl answers the door. Is the chance that both are girls one third, as the classic version says, or has meeting her changed something?
Lily Pads on a Pond
Twenty-seven lily pads, each one square foot, sit on a six-thousand-square-foot pond, and every pad doubles in area each day. How many whole days until the pads blanket the pond?
Two Bullets, One Spin
A revolver is loaded with two bullets in adjacent chambers, spun once, and fired on an empty click. Before the next pull, are you safer spinning again or pulling straight away?
The Three-Way Duel
You are the worst shot in a three-way duel against two deadlier rivals, and you fire first. Whom should you aim at to maximise your own survival?
Three Children, One Apple
You have three children and one apple, and want to use a fair coin to pick a winner with each child equally likely. What is your strategy?
One Extra Coin
I toss four coins and you toss five. How likely are you to end up with strictly more heads than I do?
Color Balls
A box holds n balls of n different colors. Repeatedly recolor a random ball to match another random ball. How many steps until every ball shares one color?
Five Pirates and 100 Coins
Five ranked pirates split 100 coins by a brutal voting rule, and every one of them reasons perfectly. How much does the most senior pirate dare to keep?
The Ticket Line
2n people line up for 5-dollar tickets, half paying with a five and half with a ten, and the seller starts with no change. How likely is everyone served without a holdup?
A Clock in Three Pieces
A clock falls and shatters into three pieces whose numbers sum to the same total. The arithmetic looks fine, so what could possibly go wrong?
When the Hands Overlap
Just after 3pm the faster minute hand starts chasing the hour hand. When does it finally catch up and sit exactly on top of it?
Angle at a Quarter Past Three
At 3:15 the minute hand points straight at the 3. The hour hand looks like it does too. How wide is the angle that is actually between them?
Find the Counterfeit Bag
Ten bags of coins, one of them counterfeit with coins a gram off. With a scale that reads exact weight, how much can a single clever weighing tell you?
The Loosened Cube Shell
A 10 by 10 by 10 cube of glued unit cubes loses its entire outer skin to weather. How many little cubes end up on the ground?
Two Missing Integers
You hold 98 of the integers from 1 to 100. Two are missing. What totals can you keep as you read the numbers once to pin down exactly which two?
Connecting Noodles
A hundred noodles, two hundred loose ends, tied together in random pairs until none remain. How many loops should you expect?
Random Ants
Five hundred ants scramble, collide, and reverse on a one-foot string. How long until the last one falls off?
Russian Roulette
One bullet, six chambers, a single spin, two players alternating. Choose to go first or second, and weigh whether the order matters.
Coin Toss Game
A and B alternate fair tosses, and whoever throws a tail right after a head wins. Does going first help?
Birthday Line
Pick a spot in a long line to be the first birthday match and win a free ticket. Which position should you choose?
Dart Game
Three darts thrown with constant skill. Given the second lands worse than the first, how likely is the third to beat both?
All-Girl World?
Every couple has children until a girl arrives, then stops. Does the stopping rule skew the fraction of girls in the world?
Boys and Girls
A two-child mother is invited to a dinner for mothers with at least one son. How likely is it that both her children are boys?
Rainbow Hats
Seven prisoners, seven hat colours, each sees the others but not his own. Can they guarantee that at least one guesses right?
Risk & Reward18
Casino Card Game
A casino deals a 52-card deck in pairs, sending black-black pairs to the dealer, red-red pairs to you, and discarding the mixed pairs. You win $100 only if your pile ends up strictly larger. What is that bet worth, and how much should you pay to play?
Roll Until Not a One
Roll a fair die until something other than a one shows up, and you are paid that face value in dollars. The ones cost only time, so what is this game worth?
The Exchange Paradox
Two sealed envelopes, one holding twice the other, and you hold one. A quick calculation seems to prove that switching raises your expected money by a quarter. Should you switch, and where does the argument go wrong?
The Marble Game
Two players each hold a red and a blue marble and reveal one at once. Matches pay A, mismatches pay B, all from an outside bank. Would you rather be A or B?
The Parlay Card
A parlay pays only if you call all four matches correctly. At 10-to-1 the payoff looks generous, but is it enough, and what about 25-to-1?
A Five-Section Wheel
A five-section wheel pays $1 on four sections and $5 on the fifth, and a spin costs $1.50. Should you play if you can spin as often as you like, and should you play if you get exactly one spin?
Three or Two
Down by two with seconds left, you can take a 40 percent three to win outright or a 70 percent two to force overtime. Which shot gives your team the better chance?
Pricing a Die Roll
A game pays a dollar for every dot on a single roll of a fair die. What ticket price makes the game fair?
Put-Call Parity
A European call and put share a strike and an expiry on a non-dividend stock. What exact relationship ties their prices together?
Never Exercise the American Call
An American call on a non-dividend stock can be exercised any time before expiry. Why is it never worth doing so early?
Black-Scholes Assumptions
The Black-Scholes formula prices an option from a handful of idealized assumptions. What are they?
Three Rolls to Stop
Roll a die up to three times, taking a face value or rolling on, but forfeiting whatever you pass up. What is the game worth and when should you stop?
Scaling Volatility with Horizon
Annual log-return volatility is 10 percent. What is the volatility of the four-year return, and why does the horizon enter the way it does?
The Dynamic Card Game
A dealer turns over a shuffled deck. Each red pays you a dollar, each black costs you one, and you may stop whenever you like. What is the game worth?
The Dynamic Dice Game
Roll a die again and again, banking each face from 1 to 5, but a 6 wipes you out. You may cash in any time. What is the game worth to a risk-neutral player?
Joint Default
Bond A defaults with probability 0.5 and bond B with 0.3, but their dependence is unknown. What are the possible ranges for the chance at least one defaults and for their correlation?
Optimal Hedge Ratio
You are long one share of A and want to short B to cancel its risk. How many shares of B minimize the variance of the hedged position?
A Dice Game
Roll a die for its face value in dollars, and roll again whenever you score a 4, 5, or 6. What payoff should you expect?
Pure Mathematics2
Sum of the First n Integers
Adding 1 through 100 one term at a time is slow. What trick collapses the whole sum, and the general sum to n, into a single line?
The Irrationality of the Square Root of 2
Could the square root of two be written as a ratio of whole numbers? A staple test of proof by contradiction.