(e. g. If an opponent owns Boardwalk with a hotel on it,
how often will you land on it?)
gives tables and graphs showing the probability
that your game piece (token) will end a turn on any given
board space and also the mean number of times you will visit
("land on") a given board space per turn. The "How to
link gives a generalized overview of the
Rules of the game:
For the complete
rules of the game of Monopoly, the reader should consult the
rulebook that comes with each set. We will briefly review the
rules that involve game piece (token) movement.
A player rolls a pair of
dice and moves his board piece (token) clockwise around the
board. The number of board spaces he moves is equal to the sum
of the dice. If he stops on a "Chance" or "Community Chest"
board space, he picks up a card from the indicated stack, and
if instructed, moves his token to a new location. If he had
doubles, he repeats this process. However, if he has three
doubles in a row, he instead goes directly to Jail.
If a player is instructed to go
to Jail (at any point during his turn), his turn ends
regardless of the doubles status.
The rules for getting out of Jail are somewhat
ambiguous in the official Monopoly booklet. If a player pays
to come out of Jail, the official rules do not define if he
may use the regular repeat dice roll if he rolls doubles, or
is restricted to a single dice roll. I have checked with
Ken Koury (Ken's
), and Rob Pratt. Both sources indicate the
"Get out of Jail" rules shown below should be used. Thus, the
statistics data shown here reflect these rules.
(Note: Rob Pratt was formerly in the Operations Research
Department at the University of North Carolina. His results for “visits per dice roll” can be
seen at http://web.archive.org/web/20030113200512/http://www.unc.edu/~rpratt/monopoly/LJRIJtable.html.
Both of us independently calculated the “visits per dice
roll” data and both of us had identical results.)
First turn in Jail: If a player wishes to get out of Jail, he
may do so by paying the $50 fine (or turning in his "Get out
of Jail Free" card) before rolling the dice. This in effect
changes his status from "In Jail" to "Just Visiting". Then he
begins his turn by rolling the dice and moving forward the
indicated number of spaces. If he had doubles, then he rolls
the dice again as per the ordinary "doubles" rule.
Alternately, the player may choose to try to stay in Jail by
not paying the $50 fine. He then rolls the dice. If the result
is doubles, he gets out of Jail for free and moves forward the
indicated number of spaces. In this instance, his turn ends
(after a possible "Chance") and he does not roll the dice
again. If the result of the dice roll was not "doubles", then
the player's turn ends, and he remains in Jail.
2nd turn in Jail: This is similar to the first turn in Jail.
If the player pays to get out, his status becomes "Just
Visiting" and he has repeat dice-roll privileges as above. If
he chooses to stay in Jail, he rolls the dice and either stays
in Jail or gets out on doubles. Again, if he gets out on
"doubles" his turn ends after a single dice roll.
3rd turn in Jail: Here the player just rolls the dice once and
moves forward the indicated number of spaces. If the dice roll
was "doubles" he gets out for free. Otherwise he must pay the
$50 fine. In either case, his turn ends after the single dice
tables define the Jail status as follows.
1) If a player plans to pay to come out on his next available
turn (first time that he can roll the dice
or is forced to come out because it is his
3rd Jail turn, then his "State" is "30". (There is a
mathematical difference between paying to come out and forced
to come out, but this is taken into consideration in the
calculations.) His "State" at the end of his turn is whatever
board space his token is on.
2) If a player wants to stay in Jail this turn (next available dice roll)
but pay to
come out the following turn (or is forced to by the 3-turn
rule), then his "State" is "40". If the subsequent dice roll
is "doubles", then he comes out and his new state becomes
whatever space his token is on when his turn ends. Else his
state changes to "30" and he stays in Jail.
3) If a player wants to stay in Jail until his 3rd turn, then
his initial "State" is "41". If he rolls doubles and gets out,
his new "State" is whatever board space his token is on. Else
he remains in Jail, and his new "State" becomes "40".
There are 16 "Chance"
cards. 10 of these instruct the player to move his token
somewhere. The calculations are based on this standard pack.
However, this has changed over the years, and possibly with
various editions of the game. The calculations are valid for
only this standard deck. (Note to manufacturers: If there are
other versions and you would be interested in new
calculations, please contact me as it is relatively easy to
make the necessary changes.)
There are 16 "Community Chest"
Two of these instruct the player to go somewhere. Any
variations here would be subject to the same rules as the
"Get out of Jail Free"
and "Community Chest" have "Get out of Jail Free" cards, which
a player may keep until used. The calculations are based on
these cards remaining in the deck. If any player holds them,
it reduces the "non move" cards in the deck. The result is a
decreased probability of remaining on the "Chance" (or
Community Chest) space at the end of a turn, and an increased
probability that you will end up on one of the "Go to" spaces.
There are also minor changes to the frequencies for other
In an actual game, the "Chance" and "Community
Chest" cards are placed on the board and sequentially accessed
during the course of a game. Unfortunately, each permutation
of the stack slightly alters the calculated frequencies for
board spaces. The only alternative is to assume that these
cards are always randomized before a player picks up a card.
The upside is: this allows a standard calculation. The
downside is: this generates the possibility that the same card
may be picked up on two consecutive plays. (Note: It is
possible to land on all three "Chance" spaces in a single
Number of ways to "Go To Jail":
the curiosities that can be counted during the computer
program search of all possible combinatorial moves is the
number of ways that you can "Go to Jail". If your game
strategy is to pay to come out of Jail at your first
opportunity, then there are 50,082 different ways to go to
Jail in one turn.
If your game strategy is to pay to get out Jail
on your 2nd turn, then there are 50,082 + 1 = 50,083 different
ways to go to Jail in one turn. When you decide to stay in
Jail on your first turn, you still might roll double 6's, land
on Chance, and be sent back to Jail.
If your game strategy is to stay in Jail until
your third turn, then there are 1 + 1 + 7 + 48,047 = 48,056
possible ways to go to Jail in one turn. There are 48,047 ways
to go to Jail assuming you don't start in Jail. If you do
start "In Jail", then both your first and 2nd turns in Jail
allow the possibility of rolling double 6's -> Chance ->
Jail as above. Finally, when you are forced to come out on
your third turn, your dice roll can total 7 in six different
ways, or you could again roll 12 -> Chance -> Jail.
of the Results:
If you search the Internet for
Monopoly probabilities, you will find different results at
various websites. The question that arises is: Which numbers
are correct (if any)? While there are no 100% guarantees, the
results presented here should be reliable for the following
Any probability results that are obtained by
simulations will always have “sampling errors”. You can reduce
these sampling errors by increasing the size of the simulation
run. Every time you multiply the sample size (number of
simulated turns) by 4, you cut the expected error factor by 2.
However, the expected error factor can never be eliminated.
The results presented here were cross checked with simulations
(hundreds of millions of simulated random dice rolls) just to
see if the results were “statistically close” to the
Any change in the Monopoly rules will produce
changes in the results. For example, if you use a different
interpretation of the “doubles” rule when coming out of Jail,
you will legitimately get different results.
There are two ways that the probabilities can be
calculated. You can calculate via “probabilities per
dice-roll”, or you can calculate via “probabilities per
player-turn”. The “per dice-roll” calculation is the easier of
the two, but the results of the two different methods can be
compared to gain additional information (e.g. calculate the
mean number of dice-rolls per turn), and to cross check for
As stated earlier, the results obtained by Rob
Pratt (See http://web.archive.org/web/20030113200512/http://www.unc.edu/~rpratt/monopoly/LJRIJtable.html
and the “per dice roll” obtained by the author are identical.
(Except that Rob’s are expressed as percents while the results
shown here are straight probabilities.) If two independent
sources use independent methods and obtain identical results,
there is a very high probability that the common result is
Next, compare the “visits per turn” with the
visits per dice roll results. If for any “non move” board
space (any board space except “Chance” and “Community Chest”),
you divide the “visits per turn” number by the “visits per
dice roll”, the number you get is the mean number of dice
rolls per turn. If your game strategy is to get out of Jail at
your first opportunity, this number is 1.1866239585+ dice
rolls per turn. If you intend to stay in Jail for as long as
possible, then the number is 1.1658963640 dice rolls per turn.
These numbers will stay the same whether you use the “Go”
space, “States Ave.”, Boardwalk, or any other space that does
not introduce intermediate moves. (Chance and Com. Chest have
different constants as the “per dice roll” data does not
include intermediate stops on the Chance and Com. Chest
spaces.) It is highly unlikely that this consistency would
exist if there were errors anywhere in the calculations.
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