Mathematics of keno and lotteries

339 Pages·2018·16.05 MB·English

by  Bollman

Mathematics of Keno
and Lotteries

Mathematics of Keno
and Lotteries
By
Mark Bollman
Albion College
Albion, Michigan, USA
CRC Press
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Version Date: 20180308
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Contents
Preface ix
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . x
1 Historical Background 1
1.1 History of Keno . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 History of Lotteries . . . . . . . . . . . . . . . . . . . . . . . 9
2 Mathematical Foundations 17
2.1 Elementary Probability . . . . . . . . . . . . . . . . . . . . . 17
2.2 Addition Rules . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Conditional Probability . . . . . . . . . . . . . . . . . . . . . 24
2.4 Random Variables and Expected Value . . . . . . . . . . . . 27
2.5 Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.6 Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . 50
2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3 Keno 59
3.1 Standard Keno Wagers . . . . . . . . . . . . . . . . . . . . . 59
3.2 Game Variations . . . . . . . . . . . . . . . . . . . . . . . . . 116
3.3 The Big Picture . . . . . . . . . . . . . . . . . . . . . . . . . 153
3.4 Video Keno . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
3.5 Keno Side Bets . . . . . . . . . . . . . . . . . . . . . . . . . . 168
3.6 Keno Strategies: Do They Work? . . . . . . . . . . . . . . . . 186
3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
4 Lotteries 203
4.1 Return of the Numbers Game . . . . . . . . . . . . . . . . . 203
4.2 Passive Lottery Tickets . . . . . . . . . . . . . . . . . . . . . 213
4.3 Lotto Games . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
4.4 Games Where Order Matters . . . . . . . . . . . . . . . . . . 264
4.5 Powerball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
4.6 Lotto Strategies: Do They Work? . . . . . . . . . . . . . . . 285
4.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
vii
viii Contents
Answers to Selected Exercises 303
References 307
Index 321
Preface
Why write a mathematics book on keno and lotteries?
Using the gambler’sadagethat “the simpler a bet is,the higher the house
advantage”, keno doesn’t seem worthy of serious attention. It’s fairly easy to
show that the house advantage (HA) in these games is among the highest in
any legitimate gambling game, often on the order of 25–35%,and the combi-
natorics involved, while certainly interesting, doesn’t appear to vary all that
much.
That’sagambler’sperspective.However,adeeperlookatthemathematics
reveals some fascinating complexity in these related and fairly simple games.
Game designers over the decades have examined “draw 20 numbers in the
range1–80”insideandout,creatingalotofvariationsthatleadtointeresting
mathematicalquestions.Acasecanbemadethatkenohaschangedmorewith
the advent of computer and video technology than any other casino game,
withtheexceptionofslotmachines,andthesechangeshaveledtomeaningful
differences in how keno can be played. For example, one popular video keno
game, Caveman Keno (page 162), cuts the HA to under 5%, and relies on
increased game speed to generate the casino’s profit.
Computerized game operations have made it possible for a wider variety
of keno games and creative wagers to reach the casino floor, and the math-
ematics involved here is sometimes more intricate and more interesting than
elementary combinatorics. A look at Penny Keno leads to an excursion into
applied mathematics where counting is only the start of the story.
Additionally, many state and provincial lotteries run keno-like games
among their offerings,which bring this game, with its history spanning many
centuries,to a wider audience. This affordsa neat transitionto lottery math-
ematics. There can be no denying the hold that multi-million dollar lotteries
haveonthe public; we needonly consider the excitementwhen the Powerball
jackpottopped$1.5billionin January2016to see thatlotteries,despite their
longoddsofwinning,canstillcapturealargeshareofmediaattention.More-
over,thereareorwereinnovativelotterygamesofferedaroundthe worldthat
aremathematicallydifferent—inaninterestingway—fromPowerballandtra-
ditional daily drawings: City Picks in Wisconsin and Lucky Lines in Oregon,
for example.
Some of the exercises included in Chapters 2–4 extend ideas explained in
the text; some of them are dedicated to exploring different games or lottery
ix