Durango Bill's
Monopoly Probabilities
The Monopoly State to State Transition Table
and
How to Calculate It
The table below shows a small portion of the Monopoly State to State Transition Table.
<--------------------
To --------------------->
Med. Community Baltic Income Reading
From Go Ave. Chest Ave. Tax Railroad
-----------------------------------------------------------------------
Go .011763 .000000 .000005 .055663 .066373 .123566
Med. Ave. .008178 .000016 .000099 .000229 .062966 .063547
Com. Chest .008389 .000000 .000007 .000016 .007068 .062019
Baltic Av. .004894 .000016 .000096 .000226 .003836 .004721
Inc. Tax .005011 .000000 .000009 .000021 .003508 .004115
etc.
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 somewhere.
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 changed.
Return to the How to Calculate the Monopoly Statistics page
Web page generated via Sea Monkey's Composer HTML editor
within a Linux Cinnamon Mint 18 operating system.
(Goodbye Microsoft)
Med. Community Baltic Income Reading
From Go Ave. Chest Ave. Tax Railroad
-----------------------------------------------------------------------
Go .011763 .000000 .000005 .055663 .066373 .123566
Med. Ave. .008178 .000016 .000099 .000229 .062966 .063547
Com. Chest .008389 .000000 .000007 .000016 .007068 .062019
Baltic Av. .004894 .000016 .000096 .000226 .003836 .004721
Inc. Tax .005011 .000000 .000009 .000021 .003508 .004115
etc.
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 somewhere.
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 changed.
Return to the How to Calculate the Monopoly Statistics page
Web page generated via Sea Monkey's Composer HTML editor
within a Linux Cinnamon Mint 18 operating system.
(Goodbye Microsoft)