Trading is deciding under uncertainty. These puzzles price bets and contracts: pot odds in poker, a call option on dice, a warranty that pays once, the value of peeking inside a box, the Kelly criterion. They teach you to turn a situation into an expected value quickly, and to notice when expected value isn't the whole story.
What these puzzles practice
Pricing a bet or a contract by expected value
Options, convexity and the value of information
Growth versus average: the Kelly criterion
19 puzzles, easiest first
Paid the square· Dice game · difficulty 1 of 5You roll a fair 6-sided die, numbered 1 to 6, once. You're paid the square of the number rolled, in dollars: a 3 pays $9, a 4 pays $16.…
The three-leg parlay· Sports betting · difficulty 1 of 5A sportsbook offers a $1 parlay on three games. You must pick the winner of all three, and you think each of your picks wins with…
To guess or not to guess· Exam strategy · difficulty 1 of 5In a math competition, each multiple-choice question has 4 options and exactly one is correct. A right answer scores +4, a wrong answer…
A call option on dice· Trading desk · difficulty 2 of 5A trading desk sells a contract on the total S of 2 fair six-sided dice that will be rolled tomorrow. It pays $100 for every point by…
Chasing the flush· Poker · difficulty 2 of 5It's the last card in a hand of Texas Hold'em. The pot holds $100, and your opponent bets $50. You're certain they're ahead unless the…
Chuck-a-luck· Carnival · difficulty 2 of 5At a carnival game you bet $1 on one of the numbers 1 to 6. Then 3 fair six-sided dice are rolled. If your number shows on k of the dice,…
Quoting to someone who knows· Market making · difficulty 2 of 5You make a market on a contract that pays, in dollars, the roll of a fair six-sided die. You'll buy one contract at $3 (your bid) or sell…
Shading the bid· Auctions · difficulty 2 of 5In a sealed-bid auction, an item is worth $80 to you. Your only rival bids an amount uniformly distributed between $0 and $100, which you…
The one-claim warranty· Electronics store · difficulty 2 of 5With your new phone, the store offers a 12-month protection plan for $120. The plan pays for one repair only: the first fault in the year.…
Worth the deductible· Insurance · difficulty 2 of 5There is a 20% chance your phone has an accident this year, and it never has more than one. If it does, the repair costs an amount…
A portfolio of moonshots· Startup investing · difficulty 3 of 5You invest $100,000 in each of 10 early-stage startups. Each one, independently, returns 20× with probability 5% and otherwise goes to…
Doubling down on red· Casino · difficulty 3 of 5At an American roulette wheel (38 pockets, 18 of them red), you bet on red, which pays even money: win and you gain your stake, lose and…
Heads, squared· Coin game · difficulty 3 of 5You flip a coin that lands heads with probability 0.5, independently each time, and keep flipping until the first tails. Let H be the…
Paid either way· Options · difficulty 3 of 5A stock trades at $100. Each day it moves exactly $1 up or $1 down, each with probability 1/2, independently of other days. A contract…
The St. Petersburg casino· Casino · difficulty 3 of 5A casino offers a game. It flips a fair coin until the first heads. If heads first appears on flip k, you win $2ᵏ: $2 if the first flip is…
What is a peek worth?· Game show · difficulty 3 of 5A sealed box holds one of the amounts $100, $200, …, $1,000, each equally likely. The host offers you a choice: keep the box, or take $550…
Pays if it ever touches· Trading desk · difficulty 4 of 5A stock closes today at $100. On each of the next 10 trading days its close moves exactly $1 up or $1 down, each with probability 1/2,…
The winner's curse· Acquisitions · difficulty 4 of 5You want to buy a company. Under its current owner it is worth V, which you only know is uniformly distributed between $0 and $100…
Too much of a good bet· Investing · difficulty 4 of 5You've found an even-money bet that wins 60% of the time. Every round you stake 50% of your current bankroll, so a win multiplies your…