Durango Bill's

Monopoly Probabilities

End-of-turn and "Land on" statistics/frequency analysis for

Parker Brothers Monopoly Game

Exact Numbers (no approximations, simulations, or estimates)

(See “Reliability of the Results” below)

(e. g. If an opponent owns Boardwalk with a hotel on it, how often will you land on it?)

Monopoly Probabilities

End-of-turn and "Land on" statistics/frequency analysis for

Parker Brothers Monopoly Game

Exact Numbers (no approximations, simulations, or estimates)

(See “Reliability of the Results” below)

(e. g. If an opponent owns Boardwalk with a hotel on it, how often will you land on it?)

The statistics link
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
calculate" link gives a generalized overview of the algorithms
involved.

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.

Dice Rolls: 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.

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 monopoly site), 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 roll.

Statistics "States": The statistical 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 again) 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" cards. Two of these instruct the player to go somewhere. Any variations here would be subject to the same rules as the "Chance" cards.

"Get out of Jail Free" Both "Chance" 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 board spaces.

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 turn.).

Number of ways to "Go To Jail": One of 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.

Reliability 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 reasons.

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 calculated predictions.

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 errors.

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 correct.

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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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.

Dice Rolls: 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.

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 monopoly site), 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 roll.

Statistics "States": The statistical 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 again) 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" cards. Two of these instruct the player to go somewhere. Any variations here would be subject to the same rules as the "Chance" cards.

"Get out of Jail Free" Both "Chance" 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 board spaces.

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 turn.).

Number of ways to "Go To Jail": One of 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.

Reliability 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 reasons.

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 calculated predictions.

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 errors.

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 correct.

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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