E_+2010+Chapter+2 - 2 Counting Chapter Concepts...

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Chapter 2 CHIU © 2.1 Chapter Concepts Multiplication Principle: counting the combination of events Permutation Combination Conditional Reasoning Sequential Counting Partitioning, Scenario Analysis Dynamic Equation and the Pascal Triangle Counting the Complement: from which to deduce the original event Data Tree: organizing a multi-attribute data set 2 Counting Preliminary and Historical Background B efore the Renaissance, probability was non-mathematical. It was not until the early sixteen century that Italian mathematicians began to examine the odds of various games of chance. There is no evidence of a calculus of chance in the Middle Ages. The Greeks used astragali (heel bones of hoofed animals) in chance games before the invention of dice. Kendall (1956) reported some early analyses of three dice games. Around 960 A.D. , Bishop Wibold of Cambrai correctly enumerated the 56 different un-ordered outcomes of playing three dice. The 216 ordered ways in which three (six-sided) dice can fall were listed in a Litan verse poem titled De Vetula , presumably from the 13 th century. In the 14 th century, advances in paper-making and block printing in Europe made playing cards popular among the wealthy. However, the church preached against it, thus its popularity was not realized until several hundred years later. Lotteries were used by the Roman Emperors as a means of financing government expenditures. Private lotteries also flourished, but were suppressed or declared illegal fearing unruly operators. They were later authorized to assist charities and the fine arts. The mathematical analysis on gaming in the 16 th and 17 th century mainly focused on the problems of dicing, ball games, table games and lotteries. It was not until the 18 th century that various card games were being analyzed. The possible outcomes of many games of chance (except for the use of astragali before suitable tools were available) can reasonably be assumed as equally likely. Thus the chances of winning are intuitively equated with the ratio of the number of favorable outcomes to the total number of possible outcomes. For example, the chance of landing even with a six-sided fair die is computed as 3/6 = 1/2. The method of counting is a book- keeping procedure to account for exactly the right number of possibilities, without over or under counts. This chapter contains a brief tour of various counting concepts and techniques. In addition to the usual treatment of permutation and combination, we emphasize two particularly useful concepts: partitioning and sequential reasoning. In conjunction with the concept of conditioning, these two ideas will be used frequently in subsequent chapters. When illuminating, we will use trees and Venn diagram to illustrate various counting principles and strategies. Counting is hard . Do not be discouraged. We will introduce dynamic equations (counting iteratively) to make counting simple(r), an approach to be profitably exploited often in later chapters. Astragalus
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This note was uploaded on 09/28/2010 for the course MS&E 220 taught by Professor Chiu during the Fall '10 term at Stanford.

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E_+2010+Chapter+2 - 2 Counting Chapter Concepts...

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