See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/257821108 The Mathematics of Slots: Configurations, Combinations, Probabilities Book * January 2013 CITATIONS 0 READS 2,048 1 author: Some of the authors of this publication are also working on these related projects: Metatheoretical analysis and structural alternatives of the resolving of the philosophical problem of the applicability of mathematics in natural sciences and Wigner's puzzle View project Editing/Proofreading/Commenting/Research Designing in philosophy of science/philosophy of mathematics Collaboration and co-authorship View project Catalin Barboianu University of Bucharest 27 PUBLICATIONS 5 CITATIONS SEE PROFILE All content following this page was uploaded by Catalin Barboianu on 01 March 2016. The user has requested enhancement of the downloaded file. THE MATHEMATICS OF SLOTS Configurations, Combinations, Probabilities This is a sample containing the title page, copyright page and content list. There is no electronic edition of this book. The printed edition can be ordered at minimal price at http://probability.infarom.ro/books.html . Cătălin Bărboianu INFAROM Publishing Applied Mathematics office@infarom.com http://www.infarom.com http://probability.infarom.ro ISBN 978-973-1991-40-5 Publisher: INFAROM Author: Cătălin Bărboianu Correction editor: CarolAnn Johnson Copyright © INFAROM 2013 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. 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Contents Introduction ............................................................................................ 9 History and evolution ............................................................................ 9 General description and rules ................................................................ 10 Popularity .............................................................................................. 12 Mathematics and slots ........................................................................... 13 The Configuration of Slot Machines .................................................... 17 Configuration of the display ............................................................... 18 Lines in a grid .................................................................................... 18 Particular lines in a rectangular grid ............................................... 27 Horizontal lines, vertical lines, and oblique lines ........................ 29 Triangular lines ............................................................................ 32 Trapezoidal lines .......................................................................... 34 Zigzag lines .................................................................................. 35 Double-stair lines ......................................................................... 36 Examples ...................................................................................... 40 Configuration of the reel .................................................................... 42 Distribution of the symbols ................................................................ 42 Arrangement of the symbols .............................................................. 44 How to estimate the number of stops and the symbol distribution on a reel ................................................................................................... 45 Method 1 – Statistical observation ..................................................... 46 Denominator-match method ........................................................... 50 Method 2 – Physical measurements ................................................... 58 Combinations .......................................................................................... 65 1. Combinations holding a specific symbol S .................................... 67 1.1 Exactly one instance of S ............................................................ 67 1.2 Exactly two instances of S .......................................................... 68 1.3 Exactly three instances of S ........................................................ 69 Generalization: Exactly m instances of S ........................................... 70 2. Combinations holding two specific symbols 1S and 2S ............... 72 2.1 Exactly one instance of 1S and one instance of 2S .................... 72 2.2 Exactly two instances of 1S and one instance of 2S ................... 73 Generalization: Exactly 1k instances of 1S and 2k instances of 2S .. 75 2.3 Only 1S and 2S ............................................................................ 77 3. Combinations holding at least one of two or more specific symbols 1S and 2S .................................................................................. 80 4. Combinations holding any symbol in several instances .............. 82 4.1 Any double .................................................................................. 82 4.2 Any triple .................................................................................... 83 Generalization: Any k-tuple ............................................................... 84 Probabilities ............................................................................................ 89 Slots gaming events ............................................................................. 93 The general formulas of the probability of the winning events related to one payline ............................................................................. 96 The probabilities of the winning events related to several lines ..... 107 Winning events related to several lines .............................................. 107 The inclusion-exclusion principle for probabilities ........................... 111 Approximations .................................................................................. 114 Approximating the probability of a conjunction of slots events ..... 117 Approximating the probability of a disjunction of slots events ...... 123 A0 approximation ........................................................................ 123 A1, A2, A3 approximations ......................................................... 125 Repeated bets and expected value ..................................................... 130 Expected value ................................................................................... 130 The general formulas of EV in slots ............................................... 132 Probabilities associated with repeated events in slots ........................ 141 Practical Applications and Numerical Results .................................... 147 3-reel slot machines ............................................................................. 149 1 x 3 display ....................................................................................... 150 Probabilities of winning events on the payline ............................... 150 1. Case A – the same number of stops and symbol distribution on the reels ............................................................................................... 150 1.1 Event 1E – A specific symbol three times ......................... 150 1.2 Event 2E – Any symbol three times (triple) ...................... 154 1.3 Event 3E – A specific symbol exactly twice ..................... 155 1.4 Event 4E – Any symbol exactly twice (double) ................ 158 1.5 Event 5E – A specific symbol exactly once ...................... 158 1.6 Event 6E – Any combination of two specific symbols ..... 161 1.7 Event 7E – Any combination of at least one of three specific symbols ....................................................................................... 167 1.8 A specific symbol at least twice ......................................... 168 1.9 A specific symbol at least once .......................................... 170 1.10 A specific symbol three times or another specific symbol twice ............................................................................................ 173 1.11 A specific symbol three times or another specific symbol once ............................................................................................. 179 1.12 A specific symbol three times or another specific symbol at least once ................................................................................. 184 1.13 A specific symbol three times or any combination of that symbol with another specific symbol ....................................................... 190 1.14 A specific symbol twice or another specific symbol once 195 1.15 A specific symbol twice or any combination of at least one of three other specific symbols .......................................................... 201 2. Case B – different numbers of stops and symbol distributions on the reels ............................................................................................... 209 2.1 Event 1E – A specific symbol three times ......................... 209 2.2 Event 2E – Any symbol three times (triple) ...................... 212 2.3 Event 3E – A specific symbol exactly twice ..................... 213 2.4 Event 4E – Any symbol exactly twice (double) ................ 216 2.5 Event 5E – A specific symbol exactly once ...................... 217 2.6 Event 6E – Any combination of two specific symbols ..... 221 2.7 Event 7E – Any combination of at least one of three specific symbols ....................................................................................... 221 2.8 A specific symbol at least twice ......................................... 222 2.9 A specific symbol at least once .......................................... 225 2.10 A specific symbol three times or another specific symbol twice ............................................................................................ 229 2.11 A specific symbol three times or another specific symbol once ............................................................................................. 229 2.12 A specific symbol three times or another specific symbol at least once ................................................................................. 230 2.13 A specific symbol three times or any combination of that symbol with another specific symbol ....................................................... 231 2.14 A specific symbol twice or another specific symbol once 232 2.15 A specific symbol twice or any combination of at least one of three other specific symbols .......................................................... 232 2 x 3 and 3 x 3 display ....................................................................... 234 5-reel slot machines ............................................................................. 235 1 x 5 display ....................................................................................... 236 Probabilities of winning events on the payline ............................... 236 1. Case A – the same number of stops and symbol distribution on the reels ............................................................................................... 236 1.1 Event 1E – A specific symbol five times ........................... 236 1.2 Event 2E – Any symbol five times (quintet) ..................... 239 1.3 Event 3E – A specific symbol exactly four times ............. 239 1.4 Event 4E – Any symbol exactly four times (quadruple) ... 242 1.5 Event 5E – A specific symbol exactly three times ............ 242 1.6 Event 6E – Any symbol exactly three times (triple) ......... 245 1.7 Event 7E – Any combination of two specific symbols ..... 245 1.8 Event 8E – Any combination of at least one of three specific symbols ....................................................................................... 251 1.9 A specific symbol at least four times ................................. 251 1.10 A specific symbol at least three times .............................. 254 1.11 A specific symbol five times or another specific symbol four times ................................................................................................. 256 1.12 A specific symbol five times or another specific symbol three times ................................................................................................ 262 1.13 A specific symbol five times or another specific symbol at least four times ..................................................................................... 267 1.14 A specific symbol five times or any combination of that symbol with another specific symbol ....................................................... 272 1.15 A specific symbol four times or any combination of at least one of three other specific symbols ................................................. 278 2. Case B – different numbers of stops and symbol distributions on the reels 285 2.1 Event 1E – A specific symbol five times ........................... 285 2.2 Event 2E – Any symbol five times (quintet) ..................... 287 2.3 Event 3E – A specific symbol exactly four times ............. 288 2.4 Event 4E – Any symbol exactly four times (quadruple) ... 290 2.5 Event 5E – A specific symbol exactly three times ............ 291 2.6 Event 6E – Any symbol exactly three times (triple) ......... 293 2.7 Event 7E – Any combination of two specific symbols ..... 294 2.8 Event 8E – Any combination of at least one of three specific symbols ....................................................................................... 294 2.9 A specific symbol at least four times ................................. 295 2.10 A specific symbol at least three times .............................. 297 2.11 A specific symbol five times or another specific symbol four times ................................................................................................. 300 2.12 A specific symbol five times or another specific symbol three times ................................................................................................ 300 2.13 A specific symbol five times or another specific symbol at least four times ..................................................................................... 301 2.14 A specific symbol five times or any combination of that symbol with another specific symbol ....................................................... 301 2.15 A specific symbol four times or any combination of at least one of three other specific symbols ................................................. 301 2 x 5, 3 x 5, 4 x 5, 5 x 5 display ........................................................ 302 9-reel slot machines ............................................................................. 304 Probabilities of winning events on several paylines .......................... 305 1. Case A – the same number of stops and symbol distribution on the reels .................................................................................................... 305 1.1 A winning event on any of the horizontal lines .................... 305 1.2 A winning event on any of the vertical lines ........................ 305 1.3 A winning event on any of the horizontal or vertical lines ... 306 1.4 A winning event on either or both of the diagonal lines ....... 306 1.5 A winning event on any of the horizontal or diagonal lines . 307 1.6 A winning event on any of the vertical or diagonal lines ..... 307 1.7 A winning event on any of the horizontal, vertical, or diagonal lines ........................................................................................... 307 2. Case B – different numbers of stops and symbol distributions on the reel ................................................................................................. 311 2.1 A winning event on any of the horizontal lines .................... 312 2.2 A winning event on any of the vertical lines ........................ 312 2.3 A winning event on any of the horizontal or vertical lines ... 312 2.4 A winning event on either or both of the diagonal lines ....... 313 2.5 A winning event on any of the horizontal or diagonal lines . 313 2.6 A winning event on any of the vertical or diagonal lines ..... 313 2.7 A winning event on any of the horizontal, vertical, or diagonal lines ........................................................................................... 314 16-reel slot machines ........................................................................... 316 Probabilities of winning events on one payline ................................. 317 1. Case A – the same number of stops and symbol distribution on the reels .................................................................................................... 317 1.1 Event 1E – A specific symbol four times ............................. 317 1.2 Event 2E – Any symbol four times (quadruple) ................... 320 1.3 Event 3E – A specific symbol exactly three times .............. 320 1.4 Event 4E – Any symbol exactly three times (triple) ............ 323 1.5 Event 5E – Any combination of two specific symbols ........ 323 1.6 Event 6E – Any combination of at least one of three specific symbols ....................................................................................... 329 1.7 A specific symbol at least three times ................................... 329 1.8 A specific symbol four times or another specific symbol three times ................................................................................................ 332 1.9 A specific symbol four times or another specific symbol at least three times ........................................................................................ 337 1.10 A specific symbol four times or any combination of that symbol with another specific symbol ....................................................... 342 1.11 A specific symbol three times or any combination of at least one of three other specific symbols ................................................. 348 2. Case B – different numbers of stops and symbol distributions on the reel ................................................................................................. 355 2.1 Event 1E – A specific symbol four times ............................. 355 2.2 Event 2E – Any symbol four times (quadruple) ................... 355 2.3 Event 3E – A specific symbol exactly three times .............. 355 2.4 Event 4E – Any symbol exactly three times (triple) ............ 355 2.5 Event 5E – Any combination of two specific symbols ........ 356 2.6 Event 6E – Any combination of at least one of three specific symbols ....................................................................................... 356 2.7 A specific symbol at least three times ................................... 356 2.8 A specific symbol four times or another specific symbol three times ................................................................................................ 356 2.9 A specific symbol four times or another specific symbol at least three times ........................................................................................ 357 2.10 A specific symbol four times or any combination of that symbol with another specific symbol ....................................................... 357 2.11 A specific symbol three times or any combination of at least one of three other specific symbols ................................................. 357 Probabilities of winning events on several paylines .......................... 358 1. Case A – the same number of stops and symbol distribution on the reels .................................................................................................... 358 1.1 A winning event on any of the horizontal lines .................... 358 1.2 A winning event on any of the vertical lines ........................ 358 1.3 A winning event on any of the horizontal or vertical lines ... 358 1.4 A winning event on either or both of the diagonals .............. 358 1.5 A winning event on any of the horizontal or diagonal lines . 359 1.6 A winning event on any of the vertical or diagonal lines ..... 359 1.7 A winning event on any of the horizontal, vertical, or diagonal lines ........................................................................................... 359 1.8 A winning event on any of the left-right trapezoidal lines .... 359 1.9 A winning event on any of the horizontal or left-right trapezoidal lines ....................................................................................... 359 References ................................................................................................ 365 View publication stats