Monopoly

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.

Main Conclusions

1. Squares do not have the same probability of being landed on.
2. 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)

Figures and Tables

Figure 1

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

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

Future Plans

I would like to simulate more games and introduce money, multiple players, and bankruptcy. Feedback greatly appreciated.