Durango Bill's
Monopoly Probabilities
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 turn?
Mean Number of Visits per Turn
What is the expected number of times you will land on various board spaces per turn?
(Click here for end-of-dice-roll probabilities instead of per turn probabilities)
The first table
shown below gives the probability that your board piece
(token) will be on any given board space at the end of a turn.
The sum of these probabilities will equal 1.000000 for each
column except for round-off error. The second table shows the
mean number of times you will visit (land on) a given board
space per turn. These columns total to more than 1.0000 as you
can visit multiple spaces in one turn via rolling doubles and
Chance/Community Chest instructions to move your game piece to
a new location.
Each table has three columns depending on your intent while in Jail. If you intend to come out of Jail at your first opportunity, it reduces the total time (probability) that you are in Jail and correspondingly increases the time (probability) you will be somewhere else.
If you choose to spend your first round in Jail and come out on your 2nd turn, one of two things might happen. If you roll doubles on your first turn, you immediately get out of Jail. If you do not roll doubles, then you stay in Jail another round. Then your status shifts to a "first round intent". However, the extra time in Jail increases your "In Jail Total" and reduces the frequency for most other board spaces. (The probability of ending your turn on "Electric Company" actually increases as you might roll double one's on your first turn.)
If you choose to spend your first two turns in Jail and come out on your 3rd turn, the results are similar to the above. You might roll doubles and come out prematurely, or you may progress through the stages described above. Your "Total Time in Jail” increases again at the expense of most other board spaces. However, once again the probability of residing on "Electric Company" increases.
The second table shows the average number of times that you will land on (visit) any particular board space during your turn. These columns total to more than 1.0000 since you may visit several board spaces in one turn. The three columns show the expectations depending on your strategy when in Jail.
The calculations assume that the "Get out of Jail Free" cards are still included in the "Chance" and "Community Chest" stacks. If some player is holding these, it reduces the probability your game piece will end the turn on the "Chance" or "Community Chest" board spaces. The probability that you will be on one of the "Go To" spaces is correspondingly increased. There are also minor changes to the probabilities for other spaces, but for the most part they are within +/- 0.5% of the values shown in the table.
Calculations assume the "Chance" and "Community Chest" cards are randomized before you draw a card from either of them. Unfortunately, the results would be different for each permutation of these cards, and the best compromise for calculation purposes is to assume they are continuously randomized.
The probabilities given in the tables can also be seen via links to the End-of-turn Probability Graph and the Visit Probability Graph. There are several items of note in the End-of-turn graph. Low values show up on the "Chance" spaces as many of them contain instructions to move to some other board space. The other end of these moves show up as local peaks. Also, if you intend to stay in Jail, the even numbered spaces at 12, 14, 16, etc. have local peaks due to the chance of rolling doubles.
Finally, these end results were tested for accuracy by simulating 100,000,000 dice rolls (for each of the three jail intents). The results very closely matched the predicted values for both for the averages and the statistical variance. The confirmation of this large sample test (run after the mathematical/computer programs results were calculated) strongly supports the notion that the results given here are accurate and exact.
Intend one Intend two Intend three
turn in Jail turns in Jail turns in Jail
End of Turn End of Turn End of Turn
State Name State Board Loc. Board Loc. Board Loc.
(Board Position) Number Probability Probability Probability
-----------------------------------------------------------------------
Mediterranean Ave. 1 0.020520013 0.019716644 0.019008868
Community Chest 2 0.019177431 0.018429526 0.017769476
Baltic Ave. 3 0.021038162 0.020217914 0.019495065
Income Tax 4 0.023085654 0.022189571 0.021398611
Reading Railroad 5 0.029179283 0.028115743 0.027220462
Oriental Ave. 6 0.022685735 0.021809680 0.021036046
Chance 7 0.008523103 0.008192667 0.007901184
Vermont Ave. 8 0.023462684 0.022557608 0.021758483
Connecticut Ave. 9 0.023154033 0.022258093 0.021467596
Jail - Just Visiting 10 0.022382528 0.021521565 0.020761241
St. Charles Place 11 0.026903674 0.025934203 0.025121658
Electric Company 12 0.024454997 0.024766564 0.025725937
States Ave. 13 0.023936064 0.022596738 0.021458554
Virginia Ave. 14 0.023534918 0.023466353 0.024134399
Pennsylvania Railroad 15 0.029773625 0.027849518 0.026232234
St. James Place 16 0.027237461 0.026643661 0.026887615
Community Chest 17 0.026809201 0.024793978 0.023116424
Tennessee Ave. 18 0.028787362 0.028162723 0.028374431
New York Ave. 19 0.031772170 0.029958722 0.028464972
Free Parking 20 0.027954295 0.027708033 0.028214957
Kentucky Ave. 21 0.028494082 0.027067217 0.025834187
Chance 22 0.010020564 0.010050858 0.010337468
Indiana Ave. 23 0.027128837 0.026079314 0.025143148
Illinois Ave. 24 0.032086565 0.030688428 0.029513693
B. & O. Railroad 25 0.030228292 0.029172073 0.028318849
Atlantic Ave. 26 0.027193862 0.025980369 0.024917838
Ventnor Ave. 27 0.026216741 0.025231604 0.024349124
Water Works 28 0.027903864 0.026822542 0.025910715
Marvin Gardens 29 0.025195820 0.024275391 0.023449158
In Jail - out next turn 30 0.046871357 0.037518952 0.030061924
Pacific Ave. 31 0.026288120 0.025296475 0.024412425
North Carolina Ave. 32 0.027088044 0.026007288 0.025050772
Community Chest 33 0.023157106 0.022252614 0.021451175
Pennsylvania Ave. 34 0.025689451 0.024671290 0.023770103
Short Line Railroad 35 0.023680196 0.022748932 0.021925619
Chance 36 0.008615329 0.008278326 0.007979819
Park Place 37 0.020861354 0.020046127 0.019325855
Luxury Tax 38 0.022154406 0.021289943 0.020525266
Boardwalk 39 0.025395658 0.024476834 0.023709142
Go (Collect $200) 40 0.031357959 0.030133174 0.029102027
In Jail - out 2nd turn 41 0.045022743 0.036074309
In Jail - out 3rd turn 42 0.043289171
"In Jail" Totals 0.046871357 0.082541695 0.109425404
Intend one Intend two Intend three
turn in Jail turns in Jail turns in Jail
Mean Number Mean Number Mean Number
State Name State of Visits of Visits of Visits
(Board Position) Number per Turn per Turn per Turn
-----------------------------------------------------------------------
Mediterranean Ave. 1 0.025291434 0.024304775 0.023434852
Community Chest 2 0.025561643 0.024563862 0.023684454
Baltic Ave. 3 0.025659582 0.024661506 0.023781539
Income Tax 4 0.027630812 0.026556798 0.025610004
Reading Railroad 5 0.035160892 0.033858469 0.032702927
Oriental Ave. 6 0.026843091 0.025803994 0.024886924
Chance 7 0.027372972 0.026314712 0.025381178
Vermont Ave. 8 0.027541069 0.026476632 0.025537130
Connecticut Ave. 9 0.027296326 0.026242986 0.025313675
Jail - Just Visiting 10 0.026930894 0.025892650 0.024976159
St. Charles Place 11 0.032058519 0.030884870 0.029841725
Electric Company 12 0.030900133 0.030752086 0.030493815
States Ave. 13 0.028147792 0.026648770 0.025369908
Virginia Ave. 14 0.029248957 0.028778814 0.028276140
Pennsylvania Railroad 15 0.034649056 0.032528690 0.030739706
St. James Place 16 0.033135487 0.032151752 0.031233415
Community Chest 17 0.035184619 0.032676165 0.030581540
Tennessee Ave. 18 0.034834360 0.033822752 0.032874594
New York Ave. 19 0.036609353 0.034542078 0.032779902
Free Parking 20 0.034217504 0.033585655 0.032934209
Kentucky Ave. 21 0.033650794 0.031947437 0.030479030
Chance 22 0.033163226 0.032871209 0.032488199
Indiana Ave. 23 0.032462303 0.031118294 0.029931816
Illinois Ave. 24 0.037803066 0.036272284 0.034924333
B. & O. Railroad 25 0.036380760 0.034978728 0.033727603
Atlantic Ave. 26 0.032124329 0.030786034 0.029614748
Ventnor Ave. 27 0.031787967 0.030506764 0.029371277
Water Works 28 0.033313499 0.032060316 0.030951659
Marvin Gardens 29 0.030686676 0.029493795 0.028432976
In Jail - out next turn 30 0.046871357 0.037518952 0.030061924
Pacific Ave. 31 0.031770320 0.030534597 0.029437893
North Carolina Ave. 32 0.031150932 0.029941035 0.028878336
Community Chest 33 0.032087056 0.030821858 0.029702902
Pennsylvania Ave. 34 0.029673051 0.028508883 0.027487591
Short Line Railroad 35 0.028866265 0.027731313 0.026727878
Chance 36 0.027430738 0.026356583 0.025414844
Park Place 37 0.025944318 0.024931451 0.024036343
Luxury Tax 38 0.025866655 0.024856519 0.023968981
Boardwalk 39 0.031160310 0.030009223 0.028985586
Go (Collect $200) 40 0.036739338 0.035295585 0.034023916
In Jail - out 2nd turn 41 0.045022743 0.036074309
In Jail - out 3rd turn 42 0.043289171
Column Totals 1.253207459 1.242611617 1.228445115
"In Jail" Totals 0.046871357 0.082541695 0.109425404
Return to the Monopoly main page
Web page generated via Sea Monkey's Composer HTML editor
within a Linux Cinnamon Mint 18 operating system.
(Goodbye Microsoft)
Each table has three columns depending on your intent while in Jail. If you intend to come out of Jail at your first opportunity, it reduces the total time (probability) that you are in Jail and correspondingly increases the time (probability) you will be somewhere else.
If you choose to spend your first round in Jail and come out on your 2nd turn, one of two things might happen. If you roll doubles on your first turn, you immediately get out of Jail. If you do not roll doubles, then you stay in Jail another round. Then your status shifts to a "first round intent". However, the extra time in Jail increases your "In Jail Total" and reduces the frequency for most other board spaces. (The probability of ending your turn on "Electric Company" actually increases as you might roll double one's on your first turn.)
If you choose to spend your first two turns in Jail and come out on your 3rd turn, the results are similar to the above. You might roll doubles and come out prematurely, or you may progress through the stages described above. Your "Total Time in Jail” increases again at the expense of most other board spaces. However, once again the probability of residing on "Electric Company" increases.
The second table shows the average number of times that you will land on (visit) any particular board space during your turn. These columns total to more than 1.0000 since you may visit several board spaces in one turn. The three columns show the expectations depending on your strategy when in Jail.
The calculations assume that the "Get out of Jail Free" cards are still included in the "Chance" and "Community Chest" stacks. If some player is holding these, it reduces the probability your game piece will end the turn on the "Chance" or "Community Chest" board spaces. The probability that you will be on one of the "Go To" spaces is correspondingly increased. There are also minor changes to the probabilities for other spaces, but for the most part they are within +/- 0.5% of the values shown in the table.
Calculations assume the "Chance" and "Community Chest" cards are randomized before you draw a card from either of them. Unfortunately, the results would be different for each permutation of these cards, and the best compromise for calculation purposes is to assume they are continuously randomized.
The probabilities given in the tables can also be seen via links to the End-of-turn Probability Graph and the Visit Probability Graph. There are several items of note in the End-of-turn graph. Low values show up on the "Chance" spaces as many of them contain instructions to move to some other board space. The other end of these moves show up as local peaks. Also, if you intend to stay in Jail, the even numbered spaces at 12, 14, 16, etc. have local peaks due to the chance of rolling doubles.
Finally, these end results were tested for accuracy by simulating 100,000,000 dice rolls (for each of the three jail intents). The results very closely matched the predicted values for both for the averages and the statistical variance. The confirmation of this large sample test (run after the mathematical/computer programs results were calculated) strongly supports the notion that the results given here are accurate and exact.
Monopoly
Board Location Probabilities
Computer Program by Bill
Butler
Intend one Intend two Intend three
turn in Jail turns in Jail turns in Jail
End of Turn End of Turn End of Turn
State Name State Board Loc. Board Loc. Board Loc.
(Board Position) Number Probability Probability Probability
-----------------------------------------------------------------------
Mediterranean Ave. 1 0.020520013 0.019716644 0.019008868
Community Chest 2 0.019177431 0.018429526 0.017769476
Baltic Ave. 3 0.021038162 0.020217914 0.019495065
Income Tax 4 0.023085654 0.022189571 0.021398611
Reading Railroad 5 0.029179283 0.028115743 0.027220462
Oriental Ave. 6 0.022685735 0.021809680 0.021036046
Chance 7 0.008523103 0.008192667 0.007901184
Vermont Ave. 8 0.023462684 0.022557608 0.021758483
Connecticut Ave. 9 0.023154033 0.022258093 0.021467596
Jail - Just Visiting 10 0.022382528 0.021521565 0.020761241
St. Charles Place 11 0.026903674 0.025934203 0.025121658
Electric Company 12 0.024454997 0.024766564 0.025725937
States Ave. 13 0.023936064 0.022596738 0.021458554
Virginia Ave. 14 0.023534918 0.023466353 0.024134399
Pennsylvania Railroad 15 0.029773625 0.027849518 0.026232234
St. James Place 16 0.027237461 0.026643661 0.026887615
Community Chest 17 0.026809201 0.024793978 0.023116424
Tennessee Ave. 18 0.028787362 0.028162723 0.028374431
New York Ave. 19 0.031772170 0.029958722 0.028464972
Free Parking 20 0.027954295 0.027708033 0.028214957
Kentucky Ave. 21 0.028494082 0.027067217 0.025834187
Chance 22 0.010020564 0.010050858 0.010337468
Indiana Ave. 23 0.027128837 0.026079314 0.025143148
Illinois Ave. 24 0.032086565 0.030688428 0.029513693
B. & O. Railroad 25 0.030228292 0.029172073 0.028318849
Atlantic Ave. 26 0.027193862 0.025980369 0.024917838
Ventnor Ave. 27 0.026216741 0.025231604 0.024349124
Water Works 28 0.027903864 0.026822542 0.025910715
Marvin Gardens 29 0.025195820 0.024275391 0.023449158
In Jail - out next turn 30 0.046871357 0.037518952 0.030061924
Pacific Ave. 31 0.026288120 0.025296475 0.024412425
North Carolina Ave. 32 0.027088044 0.026007288 0.025050772
Community Chest 33 0.023157106 0.022252614 0.021451175
Pennsylvania Ave. 34 0.025689451 0.024671290 0.023770103
Short Line Railroad 35 0.023680196 0.022748932 0.021925619
Chance 36 0.008615329 0.008278326 0.007979819
Park Place 37 0.020861354 0.020046127 0.019325855
Luxury Tax 38 0.022154406 0.021289943 0.020525266
Boardwalk 39 0.025395658 0.024476834 0.023709142
Go (Collect $200) 40 0.031357959 0.030133174 0.029102027
In Jail - out 2nd turn 41 0.045022743 0.036074309
In Jail - out 3rd turn 42 0.043289171
"In Jail" Totals 0.046871357 0.082541695 0.109425404
Monopoly Mean
Number of Visits per Turn
Computer Program by Bill Butler
Computer Program by Bill Butler
Intend one Intend two Intend three
turn in Jail turns in Jail turns in Jail
Mean Number Mean Number Mean Number
State Name State of Visits of Visits of Visits
(Board Position) Number per Turn per Turn per Turn
-----------------------------------------------------------------------
Mediterranean Ave. 1 0.025291434 0.024304775 0.023434852
Community Chest 2 0.025561643 0.024563862 0.023684454
Baltic Ave. 3 0.025659582 0.024661506 0.023781539
Income Tax 4 0.027630812 0.026556798 0.025610004
Reading Railroad 5 0.035160892 0.033858469 0.032702927
Oriental Ave. 6 0.026843091 0.025803994 0.024886924
Chance 7 0.027372972 0.026314712 0.025381178
Vermont Ave. 8 0.027541069 0.026476632 0.025537130
Connecticut Ave. 9 0.027296326 0.026242986 0.025313675
Jail - Just Visiting 10 0.026930894 0.025892650 0.024976159
St. Charles Place 11 0.032058519 0.030884870 0.029841725
Electric Company 12 0.030900133 0.030752086 0.030493815
States Ave. 13 0.028147792 0.026648770 0.025369908
Virginia Ave. 14 0.029248957 0.028778814 0.028276140
Pennsylvania Railroad 15 0.034649056 0.032528690 0.030739706
St. James Place 16 0.033135487 0.032151752 0.031233415
Community Chest 17 0.035184619 0.032676165 0.030581540
Tennessee Ave. 18 0.034834360 0.033822752 0.032874594
New York Ave. 19 0.036609353 0.034542078 0.032779902
Free Parking 20 0.034217504 0.033585655 0.032934209
Kentucky Ave. 21 0.033650794 0.031947437 0.030479030
Chance 22 0.033163226 0.032871209 0.032488199
Indiana Ave. 23 0.032462303 0.031118294 0.029931816
Illinois Ave. 24 0.037803066 0.036272284 0.034924333
B. & O. Railroad 25 0.036380760 0.034978728 0.033727603
Atlantic Ave. 26 0.032124329 0.030786034 0.029614748
Ventnor Ave. 27 0.031787967 0.030506764 0.029371277
Water Works 28 0.033313499 0.032060316 0.030951659
Marvin Gardens 29 0.030686676 0.029493795 0.028432976
In Jail - out next turn 30 0.046871357 0.037518952 0.030061924
Pacific Ave. 31 0.031770320 0.030534597 0.029437893
North Carolina Ave. 32 0.031150932 0.029941035 0.028878336
Community Chest 33 0.032087056 0.030821858 0.029702902
Pennsylvania Ave. 34 0.029673051 0.028508883 0.027487591
Short Line Railroad 35 0.028866265 0.027731313 0.026727878
Chance 36 0.027430738 0.026356583 0.025414844
Park Place 37 0.025944318 0.024931451 0.024036343
Luxury Tax 38 0.025866655 0.024856519 0.023968981
Boardwalk 39 0.031160310 0.030009223 0.028985586
Go (Collect $200) 40 0.036739338 0.035295585 0.034023916
In Jail - out 2nd turn 41 0.045022743 0.036074309
In Jail - out 3rd turn 42 0.043289171
Column Totals 1.253207459 1.242611617 1.228445115
"In Jail" Totals 0.046871357 0.082541695 0.109425404
Return to the Monopoly main page
Web page generated via Sea Monkey's Composer HTML editor
within a Linux Cinnamon Mint 18 operating system.
(Goodbye Microsoft)