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Probability and Expectation

Zun Shan
World Scientific
Publication Date: 
Number of Pages: 
Mathematical Olympiad Series 14
Problem Book
[Reviewed by
Tom Schulte
, on

The International Mathematical Olympiad (IMO), the oldest of the International Science Olympiads, is an annual contest for pre-college students. China has taken first place most of the last ten years. The author is one of the “senior coaches of China's IMO National Team” and this text is nominally a training tool. The reader does not require an IMO future to find this book useful. It is an introductory probability text that can serve as an adjunct to a more detailed text, or a collection of classroom capsules also touching on algebra, number theory, combinatorics, graph theory, and geometry.

Sixty-five short chapters make up the main content. At least to Western readers, this collection of probability problems, nicely arranged in increasing difficulty, has an exotic flavor. Inspiration and sources often come from Chinese culture: oral history and opera, legend and literature. Even Chinese dice play leads to problem setups not seen in a typical Western text. Typical card choosing setups are augmented with problems from bridge. These and more mainstream topics in introductory probability are given concise, even elegant, development, often exploring multiple approaches.

This being aimed at advanced — or at least ambitious — pre-college students, the content enters into the college level. The Bayesian coverage is as complete and effective an introduction as I have seen in any first-year college text. The biggest obstacle appears to have arisen from translation into English without sufficient follow-up editing. By my reckoning, about ten percent of the pages have grammatically difficult sentences, such as “Who is more possible to gamble away?” That same cause I am sure explains why instead of the notation \(C_k^n\) there is the form \(C_n^k\), standard in Chinese texts.

Among the numerous problems explored are single elimination competition, binomial distribution, the birthday problem, induction, derangements, balls into drawers considering distinguishable and indistinguishable scenarios (“called the pot problem”), Buffon's needle, expected value, de Méré's Problem, and more. Often, particular solutions are building blocks to general solutions for one, then two variables. 107 exercises and their answers augment and conclude this varied gallery of probability problems.

Tom Schulte is a software architect at SaaS ERP provider Plex Systems in Troy, Michigan.

  • Coin Tossing
  • General Di Qing's Coins
  • Rolling Dice
  • Wei Xiao-bao's Bet
  • Hold All the Trump Cards
  • Roll One-Spot
  • Red Balls and Black Balls
  • Same Month and Day
  • Integer Divisibility
  • Repeated Experiments
  • Silver Medal Dream
  • Fight Between Brothers
  • Subject Groups
  • More Dice
  • Custodian Turned Thief
  • Put Back or Not
  • Match Problem
  • Put Balls into Drawers
  • Problem of Matches
  • Trial in a Three-Judge Court
  • Win Twice in Succession
  • Fire Blank Shots
  • Catch a Turtle in a Jar
  • Diagnosis Rate
  • Running Well
  • Money Change Problem
  • Donkey versus Elephant
  • East Wind vesus West Wind
  • Dowry Problem (I)
  • Dowry Problem (II)
  • Job Interview
  • Boxing Match (I)
  • Boxing Match (II)
  • Boxing Match (III)
  • Rein in on the Brink of the Precipice (I)
  • Rein in on the Brink of the Precipice (II)
  • Who Will Gamble Away?
  • Equal in Strength
  • Put All Money in One Bet
  • Indeterminate Equation
  • Throw Copper Coin onto a Small Table
  • Appointment of People in Hurry
  • Obtuse Triangle
  • Buffon's Needle
  • Bertrand's paradox
  • Odd or Even Number
  • Rational or Irrational Number
  • Real Roots or Not
  • Divide the Stake
  • Sleeping Beauty
  • Number of Hits
  • The Suicide Club
  • The First Ace
  • How Many Pairs in Average
  • Many Holidays
  • Buy Lottery Tickets
  • Not to Indulge in Gambling
  • Social Party
  • Success by Trying Once or More Times
  • The 108 Heroes
  • Who is Sick?
  • A Fallen and Broken Rod
  • Broken into Three Fragments
  • The Number of Cross Points
  • Throw a Wire Ring