Jump to content
Clubplanet Nightlife Community
Sign in to follow this  
dgmodel

Monopoly...

Recommended Posts

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. 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 Butler

Monopoly Prob. with Prob. with Prob. with Total Prob

State Name Dbls = 0 Dbls = 1 Dbls = 2 for State

------------------------------------------------------------------------

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 in 1) 0.03949976 0.00000000 0.00000000 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(Dbls status) 0.84272696 0.13551857 0.02175447 1.00000000

Intend to stay in Jail 3 turns (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

------------------------------------------------------------------------

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 in 3) 0.02578439 0.03094127 0.03712952 0.09385517

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

Totals (Dbls status)0.78963838 0.15353635 0.05682527 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

--------------------------------------------------------------

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

Share this post


Link to post
Share on other sites

Create an account or sign in to comment

You need to be a member in order to leave a comment

Create an account

Sign up for a new account in our community. It's easy!

Register a new account

Sign in

Already have an account? Sign in here.

Sign In Now

Sign in to follow this  

×