The Mathematics of Monopoly: how likely are you to land on a square?
This project is inspired by standupmaths' The Mathematics of Winning Monopoly.
Monopoly is a board game of chance in which players go around a board to buy and develop properties and pay rents. However, not all of the squares have equal probability of being landed on because of two main factors:
Chance and Community Chest cards include a mechanism of sending the player to another square or to jail Jail: A player goes to jail by rolling three doubles in a row, getting sent there from Chance or Community Chest card, or landing on the Go To Jail square.
I calculated the probabilities of landing on every square by simulating 500000 games of 100 turns each in this Jupyter notebook.
- Squares do not have the same probability of being landed on.
- The squares most likely to be landed on are in descending order:
- Jail
- The orange squares (Tennessee Avenue, New York Avenue, St. James Place)
- The red squares (Illinois Avenue, Kentucky Avenue, Indiana Avenue)
- The yellow squares (Atlantic Avenue, Ventnor Avenue, Marvin Gardens)
Figure 1 below shows the relative probabilities of moving from one square to another. As expected, the most likely squares from a given square are those 6, 7, and 8 squares away since these are the numbers that come up the most when rolling two dice (44% of the time). We also note that properties 6, 7, and 8 squares away from jail (square 40) have high visit rates since a player ends in jail a lot. . Other important squares are destinations from chance and community card squares. Note the blank row/column 30 (Go To Jail square). This is because no one ends the turn on this square.
Table 1 below shows the simulated probabilities of landing on every square
index | names | percentage |
---|---|---|
0 | Go | 2.885436 |
1 | Mediterranean Avenue | 2.003141 |
2 | Community chest | 1.918602 |
3 | Baltic Avenue | 2.082590 |
4 | Income tax | 2.272565 |
5 | Reading railroad | 2.753477 |
6 | Oriental Avenue | 2.248173 |
7 | Chance | 2.168054 |
8 | Vermont Avenue | 2.305446 |
9 | Connecticut Avenue | 2.269254 |
10 | Jail | 2.224882 |
11 | St. Charles Place | 2.629665 |
12 | Electric company | 2.663134 |
13 | States Avenue | 2.251561 |
14 | Virginia Avenue | 2.479692 |
15 | Pennysylvania Railroad | 2.609909 |
16 | St. James Place | 2.744481 |
17 | Community chest | 2.572099 |
18 | Tennessee Avenue | 2.869067 |
19 | New York Avenue | 2.867210 |
20 | Free parking | 2.846641 |
21 | Kentucky Avenue | 2.655180 |
22 | Chance | 2.630511 |
23 | Indiana Avenue | 2.596563 |
24 | Illinois Avenue | 3.023932 |
25 | B & O Railroad | 2.742805 |
26 | Atlantic Avenue | 2.578784 |
27 | Ventnor Avenue | 2.560849 |
28 | Water works | 2.686024 |
29 | Marvin Gardens | 2.480863 |
30 | Go to Jail | 0.000000 |
31 | Pacific Avenue | 2.537292 |
32 | North Carolina Avenue | 2.473935 |
33 | Community chest | 2.389051 |
34 | Pennsylvania Avenue | 2.348941 |
35 | Short line | 2.283079 |
36 | Chance | 2.034814 |
37 | Luxury tax | 2.050690 |
38 | Park Place | 2.052077 |
39 | Boardwalk | 2.471845 |
40 | Actual Jail | 3.737686 |
I would like to simulate more games and introduce money, multiple players, and bankruptcy. Feedback greatly appreciated.