THE MATHEMATICS OF LOTTERY Odds, Combinations, Systems

THE MATHEMATICS OF LOTTERY

Odds, Combinations, Systems

Ctlin Brboianu

INFAROM Publishing Applied Mathematics office@



ISBN 978-973-1991-11-5

Publisher: INFAROM Author: Ctlin Brboianu Correction Editor: CarolAnn Johnson

Copyright ? INFAROM 2009

This work is subject to copyright. All rights are reserved, whether the whole work or part of the material is concerned, specifically the rights of translation, reprinting, reuse of formulas and tables, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks.

Duplication of this publication or parts thereof is permitted only under the provisions of Copyright Laws, and permission for use must always be obtained from INFAROM.

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Contents (of the complete edition)

Introduction ...................................................................................... 5 The Rules of Lottery ....................................................................... 11 Supporting Mathematics ................................................................ 15

Probability space .......................................................................... 16 Probability properties and formulas used ..................................... 19 Combinatorics ............................................................ 22 Parameters of the lottery matrices ....................................... 25 Number Combinations ........................................................ 27 Probabilities of Winning with Simple Lines .................................. 37 General formula of the winning probability ............................ 37 Cumulated winning probabilities ....................................... 53 Intermediary probabilities .............................................. 67 Enhancing the Winning Probability ........................................ 75 Probabilities of winning with systems .................................. 76 Probability thresholds ................................................... 91 Compound Lines ................................................................. 107 Unfolding compound lines ................................................. 108 Winning probabilities ..................................................... 111 The number of prizes ................................................... 127 Systems of compound lines ........................................................... 136 Bridgehead Systems ....................................................... 141 Winning probabilities ..................................................... 142 The number of prizes ................................................... 155 Reduced Systems ............................................................. 159 Obtaining the reduced systems ......................................... 161 Winning probabilities ..................................................... 173 Probabilities of Repeated Events ......................................... 177 The Strategy of Choosing ................................................ 191 References ......................................................................................... 203

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Introduction

Lottery is by far the oldest and the most widely known game of chance, having been practiced since antiquity. In its various forms, the lottery preserves a basic structure and technical procedure that makes it the easiest and most popular game of chance: the random draw from an urn of some objects (balls, tickets, lots, plates, slips, etc.) containing predefined symbols (numbers, images, words, etc.), followed by the distribution of prizes for players who made correct predictions regarding this draw, according to some pre-established rules. Particularizing this definition, we find forms of lottery even in the simple procedures of drawing lots or organizing tombolas.

Nowadays, the most prevalent form of lottery is that with randomly selected numbers; winning categories are based on the number of numbers correctly predicted on the playing ticket. The most popular forms of these games are the national and state lotteries.

The early history of the lottery can be traced back to the second millennium B.C. In the Chinese The Book of Songs there is a reference to a game of chance known as "the drawing of wood", which in context appears to describe the drawing of lots. The first signs of a lottery trace back the Han Dynasty between 205 and 187 B.C., where ancient Keno slips (a form of lottery also practiced nowadays) were discovered. The first known European lottery occurred during the Roman Empire. The earliest record of a public lottery offering tickets for sale is the lottery organized by the Roman Emperor Augustus Caesar. Proceeds went for repairs to the city of Rome, and the winners were given prizes in the form of valuable articles.

The first recorded lotteries to offer tickets for sale with prizes in the form of money were held in the Low Countries during the period 1443?1449, and their funds were used for town fortifications. In the 17th century, it was quite common in the Netherlands to organize lotteries to collect money for the poor; the prizes were paintings. The Dutch were the first to shift the lottery to solely monetary prizes and to base prizes on odds (roughly about 1 in 4 tickets winning a

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