What is the probability your board piece (token) will be on any
given board space at the end of a turn?
What is the expected number of times you will land on various
board spaces per turn?
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.
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
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
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