dgmodel Posted August 26 Report Share Posted August 26 Monopoly Probabilities per Dice RollMonopoly Board Location Probabilities (Statistics, Frequency)What is the probability your board piece (token) will be on any given board space at the end of a dice roll?Monopoly Dice Roll ProbabilitiesWhat is the mean number of times you will land on various board spaces per dice roll? The "End of Turn" page shows the probability that your game piece will be residing on any of the Monopoly board spaces at the end of your turn. The "Visits" table shows the mean number of times you will land on a particular board space during your turn. It is also possible to calculate the probability of your game piece being on any particular board space at the end of a dice roll. (Note: You may have up to 3 dice rolls per turn due to rolling doubles.) The data shown below gives the end-of-dice-roll probability that your game piece will be on a given board space along with your current dice doubles status. Additional information on end-of-dice-roll probabilities can be found at other websites - especially those by Rob Pratt and Truman Collins. (Rob Pratt's tables show similar results to those given here.) It is interesting to compare the data shown below with that in the "Visits" table. The unit of measure in the "Visits" table is "Visits per turn". The unit of measure in the tables below is "Visits" (probability) per dice roll. If we take the value for any board space (except Chance and Community Chest) from the "Visits" table and divide by the comparable value from the tables below, we will get a constant that gives the mean number of dice rolls per turn. If your game strategy is to get out of Jail at your first opportunity, this constant is 1.18662 dice rolls per turn. If you try to stay in Jail as long as possible, it slightly reduces this value to 1.16590 dice rolls per turn. Given that your game strategy is to come out of Jail at the first opportunity, the first table below shows the steady state probability that your game piece will be on any given board space along with your dice doubles status. The second table shows similar data assuming you want to stay in Jail as long as possible. For example: if your game strategy is to come out of Jail as soon as possible, then at the end of any given dice roll, there is a .01772943 probability that your game piece will be on Baltic Ave. and your dice status is no doubles. There is also a .00327499 probability that you will be on Baltic and you have rolled exactly one doubles so far in your current turn. Finally, the "Jail" data shows your status as per the number of turns you want to stay in Jail. The "0" column shows the probability you will be coming out this dice roll (same as turn) while the "2" column shows the probability given that you want to stay in Jail as long as possible. If your intent is to stay in Jail as long as possible, then subsequent dice rolls (turns) will shift your status leftward if you don't get doubles.Intend to stay in Jail 1 turn (Choice of 1, 2, or 3)Computer program by Bill ButlerMonopoly Prob. with Prob. with Prob. with Total ProbState Name Dbls = 0 Dbls = 1 Dbls = 2 for State------------------------------------------------------------------------Mediterranean Ave. 0.01729277 0.00338444 0.00063656 0.02131377Community Chest 0.01616134 0.00235360 0.00033387 0.01884880Baltic Ave. 0.01772943 0.00327499 0.00061961 0.02162402Income Tax 0.01945490 0.00334457 0.00048576 0.02328523Reading Railroad 0.02459017 0.00426663 0.00077423 0.02963103Oriental Ave. 0.01911788 0.00304506 0.00045846 0.02262140Chance 0.00718265 0.00123490 0.00023293 0.00865048Vermont Ave. 0.01977264 0.00297474 0.00046222 0.02320960Connecticut Ave. 0.01951253 0.00293825 0.00055257 0.02300335Jail - Visiting 0.01886236 0.00332231 0.00051072 0.02269539St. Charles Place 0.02267245 0.00369691 0.00064721 0.02701658Electric Company 0.02060889 0.00483545 0.00059604 0.02604038States Ave. 0.02017157 0.00302722 0.00052211 0.02372090Virginia Ave. 0.01983351 0.00426327 0.00055210 0.02464888Pa. Railroad 0.02509104 0.00351882 0.00058984 0.02919969St. James Place 0.02295374 0.00436528 0.00060515 0.02792417Community Chest 0.02259284 0.00289772 0.00045409 0.02594465Tennessee Ave. 0.02425989 0.00446247 0.00063350 0.02935585New York Ave. 0.02677526 0.00355060 0.00052583 0.03085169Free Parking 0.02355784 0.00460530 0.00067288 0.02883601Kentucky Ave. 0.02401273 0.00380044 0.00054526 0.02835843Chance 0.00844460 0.00176642 0.00026931 0.01048033Indiana Ave. 0.02286220 0.00392544 0.00056921 0.02735686Illinois Ave. 0.02704021 0.00402884 0.00078861 0.03185766B. & O. Railroad 0.02547420 0.00451952 0.00066533 0.03065905Atlantic Ave. 0.02291700 0.00350249 0.00065254 0.02707204Ventnor Ave. 0.02209355 0.00407801 0.00061701 0.02678858Water Works 0.02351534 0.00388255 0.00067630 0.02807418Marvin Gardens 0.02123320 0.00399474 0.00063255 0.02586049In Jail (out in 1) 0.03949976 0.00000000 0.00000000 0.03949976Pacific Ave. 0.02215371 0.00395698 0.00066302 0.02677370North Carolina Ave. 0.02282782 0.00292986 0.00049404 0.02625173Community Chest 0.01951512 0.00352920 0.00061623 0.02366055Pennsylvania Ave. 0.02164919 0.00290958 0.00044750 0.02500628Short Line RR 0.01995594 0.00370367 0.00066677 0.02432638Chance 0.00726037 0.00122864 0.00017972 0.00866873Park Place 0.01758043 0.00362294 0.00066061 0.02186398Luxury Tax 0.01867012 0.00272694 0.00040148 0.02179853Boardwalk 0.02140161 0.00410866 0.00074937 0.02625963Go (Collect $200) 0.02642620 0.00394114 0.00059389 0.03096123Totals(Dbls status) 0.84272696 0.13551857 0.02175447 1.00000000Intend to stay in Jail 3 turns (1, 2, or 3)Computer program by Bill ButlerMonopoly Prob. with Prob. with Prob. with Total ProbState Name Dbls = 0 Dbls = 1 Dbls = 2 for State------------------------------------------------------------------------Mediterranean Ave. 0.01630408 0.00319910 0.00059711 0.02010029Community Chest 0.01524104 0.00221871 0.00031532 0.01777508Baltic Ave. 0.01672110 0.00309331 0.00058324 0.02039764Income Tax 0.01835379 0.00315364 0.00045851 0.02196594Reading Railroad 0.02334724 0.00397722 0.00072514 0.02804960Oriental Ave. 0.01804281 0.00287202 0.00043091 0.02134574Chance 0.00677692 0.00116760 0.00021911 0.00816363Vermont Ave. 0.01866245 0.00280688 0.00043410 0.02190343Connecticut Ave. 0.01841295 0.00277947 0.00051935 0.02171177Jail - Visiting 0.01780711 0.00313516 0.00048001 0.02142228St. Charles Place 0.02154708 0.00344689 0.00060155 0.02559552Electric Company 0.02206537 0.00352772 0.00056173 0.02615483States Ave. 0.01840520 0.00286415 0.00049066 0.02176000Virginia Ave. 0.02070030 0.00306035 0.00049206 0.02425270Pa. Railroad 0.02249963 0.00331171 0.00055439 0.02636573St. James Place 0.02306175 0.00321199 0.00051544 0.02678919Community Chest 0.01982717 0.00269765 0.00042649 0.02295131Tennessee Ave. 0.02433701 0.00334277 0.00051706 0.02819684New York Ave. 0.02441467 0.00321471 0.00048624 0.02811562Free Parking 0.02420023 0.00351761 0.00053014 0.02824797Kentucky Ave. 0.02215822 0.00347519 0.00050874 0.02614214Chance 0.00886654 0.00137679 0.00020620 0.01044953Indiana Ave. 0.02156551 0.00357922 0.00052806 0.02567279Illinois Ave. 0.02531417 0.00403949 0.00060127 0.02995492B. & O. Railroad 0.02428934 0.00403866 0.00060047 0.02892847Atlantic Ave. 0.02137226 0.00351333 0.00051525 0.02540084Ventnor Ave. 0.02088447 0.00374318 0.00056437 0.02519201Water Works 0.02222386 0.00376146 0.00056220 0.02654752Marvin Gardens 0.02011256 0.00369832 0.00057635 0.02438722In Jail (out in 3) 0.02578439 0.03094127 0.03712952 0.09385517Pacific Ave. 0.02093876 0.00370624 0.00060415 0.02524915North Carolina Ave. 0.02148628 0.00283270 0.00045024 0.02476921Community Chest 0.01839887 0.00332677 0.00056625 0.02229190Pennsylvania Ave. 0.02038783 0.00275731 0.00043122 0.02357636Short Line RR 0.01880580 0.00350526 0.00061368 0.02292475Chance 0.00684436 0.00115400 0.00017609 0.00817445Park Place 0.01657596 0.00342861 0.00061162 0.02061619Luxury Tax 0.01760471 0.00256429 0.00038941 0.02055841Boardwalk 0.02033555 0.00383074 0.00069491 0.02486120Go (Collect $200) 0.02496108 0.00366483 0.00055672 0.02918262Totals (Dbls status)0.78963838 0.15353635 0.05682527 1.00000000How to Calculate the Monopoly Probabilities per Dice Roll Calculations for the above tables are similar to those for the "end of turn" data in that a "State to State Transition Table" is initialized and the solved as a Markov chain. However, the structure of the transition table is different as it contains 120 rows and columns. The rows (and columns) are defined by the Monopoly board spaces and the current number of dice doubles. A brief section of the transition table appears below where row names are "Board space, dice doubles count". The data in the table shows the single dice roll probability of going from the "row state" to the "column state". Baltic Baltic Baltic IncTax IncTax IncTax ReadRR ReadRR Dbls=0 Dbls=1 Dbls=2 Dbls=0 Dbls=1 Dbls=2 Dbls=0 Dbls=1 --------------------------------------------------------------MedAv,0 .00000 .02778 .00000 .06250 .00174 .00000 .06250 .02951MedAv,1 .00000 .00000 .02778 .06250 .00000 .00174 .06250 .00000MedAv,2 .00000 .00000 .00000 .06250 .00000 .00000 .06250 .00000ComCh,0 .00000 .00000 .00000 .00694 .02778 .00000 .06250 .00000ComCh,1 .00000 .00000 .00000 .00694 .00000 .02778 .06250 .00000ComCh,2 .00000 .00000 .00000 .00694 .00000 .00000 .06250 .00000 Quote Link to comment Share on other sites More sharing options...
danamdkny Posted August 26 Report Share Posted August 26 Quote Link to comment Share on other sites More sharing options...
dgmodel Posted August 27 Author Report Share Posted August 27 Monopoly Board Location Probabilities (Statistics, Frequency) Quote Link to comment Share on other sites More sharing options...
georgym Posted August 27 Report Share Posted August 27 Just play for FUN by the way, im the dude on the horse Quote Link to comment Share on other sites More sharing options...
romy20 Posted August 27 Report Share Posted August 27 Monopoly's the best!!!!I used to love scrabble, but now i like monopoly better.whenever im online i always go to games.com and play it lolits really addicting actually lol Quote Link to comment Share on other sites More sharing options...
dgmodel Posted August 27 Author Report Share Posted August 27 link? Quote Link to comment Share on other sites More sharing options...
danamdkny Posted August 27 Report Share Posted August 27 Originally posted by gmccookny Just play for FUN by the way, im the dude on the horse - YEAH - - I am the TOP Hat:D Quote Link to comment Share on other sites More sharing options...
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