Durango Bill's
Monopoly Probabilities
Monopoly Probabilities per Dice Roll
Monopoly 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 Probabilities
What 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.
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 0.01772943 probability that your game piece will be on Baltic Ave. and your dice status is no doubles. There is also a 0.00327499 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. If your intent is to stay in Jail as long as possible, then subsequent dice rolls (turns) will shift your status upward to the next higher jail state if you don't get doubles.
Intend to stay in Jail 1 turn (Choice of 1, 2, or 3)
Computer program by Bill Butler
Monopoly Prob. with Prob. with Prob. with Total Prob
State Name Dbls = 0 Dbls = 1 Dbls = 2 for State
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Mediterranean Ave. 0.01729277 0.00338444 0.00063656 0.02131377
Community Chest 0.01616134 0.00235360 0.00033387 0.01884880
Baltic Ave. 0.01772943 0.00327499 0.00061961 0.02162402
Income Tax 0.01945490 0.00334457 0.00048576 0.02328523
Reading Railroad 0.02459017 0.00426663 0.00077423 0.02963103
Oriental Ave. 0.01911788 0.00304506 0.00045846 0.02262140
Chance 0.00718265 0.00123490 0.00023293 0.00865048
Vermont Ave. 0.01977264 0.00297474 0.00046222 0.02320960
Connecticut Ave. 0.01951253 0.00293825 0.00055257 0.02300335
Jail - Visiting 0.01886236 0.00332231 0.00051072 0.02269539
St. Charles Place 0.02267245 0.00369691 0.00064721 0.02701658
Electric Company 0.02060889 0.00483545 0.00059604 0.02604038
States Ave. 0.02017157 0.00302722 0.00052211 0.02372090
Virginia Ave. 0.01983351 0.00426327 0.00055210 0.02464888
Pa. Railroad 0.02509104 0.00351882 0.00058984 0.02919969
St. James Place 0.02295374 0.00436528 0.00060515 0.02792417
Community Chest 0.02259284 0.00289772 0.00045409 0.02594465
Tennessee Ave. 0.02425989 0.00446247 0.00063350 0.02935585
New York Ave. 0.02677526 0.00355060 0.00052583 0.03085169
Free Parking 0.02355784 0.00460530 0.00067288 0.02883601
Kentucky Ave. 0.02401273 0.00380044 0.00054526 0.02835843
Chance 0.00844460 0.00176642 0.00026931 0.01048033
Indiana Ave. 0.02286220 0.00392544 0.00056921 0.02735686
Illinois Ave. 0.02704021 0.00402884 0.00078861 0.03185766
B. & O. Railroad 0.02547420 0.00451952 0.00066533 0.03065905
Atlantic Ave. 0.02291700 0.00350249 0.00065254 0.02707204
Ventnor Ave. 0.02209355 0.00407801 0.00061701 0.02678858
Water Works 0.02351534 0.00388255 0.00067630 0.02807418
Marvin Gardens 0.02123320 0.00399474 0.00063255 0.02586049
In Jail - out next turn 0.03949976 0.03949976
Pacific Ave. 0.02215371 0.00395698 0.00066302 0.02677370
North Carolina Ave. 0.02282782 0.00292986 0.00049404 0.02625173
Community Chest 0.01951512 0.00352920 0.00061623 0.02366055
Pennsylvania Ave. 0.02164919 0.00290958 0.00044750 0.02500628
Short Line RR 0.01995594 0.00370367 0.00066677 0.02432638
Chance 0.00726037 0.00122864 0.00017972 0.00866873
Park Place 0.01758043 0.00362294 0.00066061 0.02186398
Luxury Tax 0.01867012 0.00272694 0.00040148 0.02179853
Boardwalk 0.02140161 0.00410866 0.00074937 0.02625963
Go (Collect $200) 0.02642620 0.00394114 0.00059389 0.03096123
Totals 0.84272696 0.13551857 0.02175447 1.00000000
Intend to stay in Jail 3 turns (Choice of 1, 2, or 3)
Computer program by Bill Butler
Monopoly Prob. with Prob. with Prob. with Total Prob
State Name Dbls = 0 Dbls = 1 Dbls = 2 for State
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Mediterranean Ave. 0.01630408 0.00319910 0.00059711 0.02010029
Community Chest 0.01524104 0.00221871 0.00031532 0.01777508
Baltic Ave. 0.01672110 0.00309331 0.00058324 0.02039764
Income Tax 0.01835379 0.00315364 0.00045851 0.02196594
Reading Railroad 0.02334724 0.00397722 0.00072514 0.02804960
Oriental Ave. 0.01804281 0.00287202 0.00043091 0.02134574
Chance 0.00677692 0.00116760 0.00021911 0.00816363
Vermont Ave. 0.01866245 0.00280688 0.00043410 0.02190343
Connecticut Ave. 0.01841295 0.00277947 0.00051935 0.02171177
Jail - Visiting 0.01780711 0.00313516 0.00048001 0.02142228
St. Charles Place 0.02154708 0.00344689 0.00060155 0.02559552
Electric Company 0.02206537 0.00352772 0.00056173 0.02615483
States Ave. 0.01840520 0.00286415 0.00049066 0.02176000
Virginia Ave. 0.02070030 0.00306035 0.00049206 0.02425270
Pa. Railroad 0.02249963 0.00331171 0.00055439 0.02636573
St. James Place 0.02306175 0.00321199 0.00051544 0.02678919
Community Chest 0.01982717 0.00269765 0.00042649 0.02295131
Tennessee Ave. 0.02433701 0.00334277 0.00051706 0.02819684
New York Ave. 0.02441467 0.00321471 0.00048624 0.02811562
Free Parking 0.02420023 0.00351761 0.00053014 0.02824797
Kentucky Ave. 0.02215822 0.00347519 0.00050874 0.02614214
Chance 0.00886654 0.00137679 0.00020620 0.01044953
Indiana Ave. 0.02156551 0.00357922 0.00052806 0.02567279
Illinois Ave. 0.02531417 0.00403949 0.00060127 0.02995492
B. & O. Railroad 0.02428934 0.00403866 0.00060047 0.02892847
Atlantic Ave. 0.02137226 0.00351333 0.00051525 0.02540084
Ventnor Ave. 0.02088447 0.00374318 0.00056437 0.02519201
Water Works 0.02222386 0.00376146 0.00056220 0.02654752
Marvin Gardens 0.02011256 0.00369832 0.00057635 0.02438722
In Jail - out next turn 0.02578439 0.02578439
Pacific Ave. 0.02093876 0.00370624 0.00060415 0.02524915
North Carolina Ave. 0.02148628 0.00283270 0.00045024 0.02476921
Community Chest 0.01839887 0.00332677 0.00056625 0.02229190
Pennsylvania Ave. 0.02038783 0.00275731 0.00043122 0.02357636
Short Line RR 0.01880580 0.00350526 0.00061368 0.02292475
Chance 0.00684436 0.00115400 0.00017609 0.00817445
Park Place 0.01657596 0.00342861 0.00061162 0.02061619
Luxury Tax 0.01760471 0.00256429 0.00038941 0.02055841
Boardwalk 0.02033555 0.00383074 0.00069491 0.02486120
Go (Collect $200) 0.02496108 0.00366483 0.00055672 0.02918262
In Jail - out 2nd turn 0.03094127 0.03094127
In Jail - out 3rd turn 0.03712952 0.03712952
Totals 0.85770917 0.12259508 0.01969575 1.00000000
How 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
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MedAv,0 .00000 .02778 .00000 .06250 .00174 .00000 .06250 .02951
MedAv,1 .00000 .00000 .02778 .06250 .00000 .00174 .06250 .00000
MedAv,2 .00000 .00000 .00000 .06250 .00000 .00000 .06250 .00000
ComCh,0 .00000 .00000 .00000 .00694 .02778 .00000 .06250 .00000
ComCh,1 .00000 .00000 .00000 .00694 .00000 .02778 .06250 .00000
ComCh,2 .00000 .00000 .00000 .00694 .00000 .00000 .06250 .00000
If you try to calculate these by hand, don't forget things like rolling doubles (If not 3rd doubles, the "doubles" status increases by 1.), landing on "Chance" (Advance to Reading, Go back 3 spaces), etc. The entire table will have 120x120 = 14,400 entries which is why it is much easier to generate the entire table via a computer program.
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