The table below shows a small portion of the
Monopoly State to State Transition Table.
.055663 .066373 .123566
Ave. .008178 .000016
.000021 .003508 .004115
Each entry shows the probability of a player's
token going from one of the spaces (states) of a Monopoly
board and ending his turn on another space (state). The entire
table is 40x40 when you intend to come out of Jail on your
first turn, and 42x42 if you intend to stay in Jail 3 turns.
For example, the probability of ending your turn
on Community Chest (Monopoly space "2") given that you started
on "Go" is .000005. Each of the above values is calculated by
generating all possible combinations starting on the "From"
space and ending on the "To" space. Then for each of these
combinations, you have to calculate the probability of this
route (multiply the probabilities for each stage), and finally
you have to add the partial results together.
For example, the .000005 value for "Go" to "Com.
Chest" is calculated as follows: (Values in parentheses show
the probability of each stage)
One possible sequence might be: From "Go",
roll double 6's (1/36), times roll double 5's (1/36) (on
Chance), times draw "Advance to Board Walk" (1/16), times roll
3 (1/18) (on Com. Chest), draw a Com. Chest card times the
probability that you stay on Com. Chest (14/16) = .000002344.
Another sequence would start with double 5's
followed by double 6's and then repeat the above sequence
which again produces a probability of .000002344. Fortunately
this completes the possible sequences. When the two
.000002344's are added together, we get .000004689 which when
rounded to 6 spaces to the right of the decimal point gives
the .000005 shown in the table.
If we calculate the state to state value for "Go"
to "Income Tax”, the number of combinations increases. You
could simply roll a "4" (not doubles), or roll 4 (doubles) -
roll again getting a 3 (on Chance) and draw "Go Back 3
Spaces". Other combinations could take you around the Monopoly
Board via "Advance to Go", "Advance to Board Walk", and on
your 3rd dice roll you end your turn on Income Tax. Just
trying to generate all possible state to state routes for some
of the combinations can get somewhat tedious. On top of this
you also have to calculate the probabilities.
"In Jail" combinations add still further
complications. If you intend to always come out of Jail at
your first opportunity, then there is only one "in Jail"
state. If you intend to stay in Jail until your second dice
roll, then you must create a second "in Jail" state. For this
second "in Jail", your dice roll would either get you out of
jail to some board space (via doubles), or you would remain in
Jail and sequence to the "First" Jail state. Similarly, if you
wish to stay in Jail until your third dice roll (requires 3
"in Jail" states), then you would sequence to the "2nd" "Jail"
state if you did not roll doubles.
The result of all the calculations will produce
the entire state to state transition table (exact size various
with the number of "Jail" states). One of the ways to check
your calculations would be to form the sum of the
probabilities for each row. The sum of all the entries on each
row MUST total exactly 1.00000 (except for slight round-off
errors). If they do not, you have made calculation errors
The treatment of the "Chance" and "Community
Chest" cards presents a problem. In an actual game, these
cards are placed on the table and then cycled through for the
remainder of the game. Unfortunately, the exact order will
cause changes in the State to State table. As an effort to
nullify this factor, the calculations used here assume these
decks are randomized before each instance of drawing a card.
This corrects the "Order" problem, but introduces the chance
of drawing the same card twice in a given turn. Another
possible way to calculate the state to state table would be to
randomize these decks before each turn, but in effect remove
each card from the deck while still within the player's turn.
This would probably be slightly closer to game realities, but
the downside is that it significantly increases the complexity
of the calculations. The overall errors introduced by the
randomize-for-each-draw vs. randomize-for-each-turn probably
tend to cancel each other and hence the easier calculation
method was used.
Finally, it should be noted that the "Get out of
Jail Free" cards are assumed to be in the respective card
decks. If some player holds these, then the probabilities for
the remaining cards in the decks are
the How to Calculate the Monopoly Statistics page
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