## Friday, May 6, 2016

### Monopoly Probability -- Question One

First Roll Results in a Game of Monopoly

Question:   Players start the game of monopoly from Go and throw two dice.   The sum of the two dice rolls ranges from 2 to 12.   List all squares that the player may land on after one roll of the two dice.   What are the likelihoods of landing on each property after one roll of the two dice?

Analysis:   There are six numbers on each dice resulting in 36 combinations of dice roll outcomes.   Not all dice throw outcomes are unique.   For instance (1,2) gives the same result as (2,1).   Also, (1,3) (3,1) and (2,2) all give the sum of 4.

In a previous post we calculated the unique outcomes from the throws of two dice.

The outcomes are listed below.

 Unique Outcomes From Two Dice Throws Outcome # Unique Pair Sum Probability 1 1 1 2 0.027777778 2 1 2 3 0.055555556 3 1 3 4 0.055555556 4 1 4 5 0.055555556 5 1 5 6 0.055555556 6 1 6 7 0.055555556 7 2 2 4 0.027777778 8 2 3 5 0.055555556 9 2 4 6 0.055555556 10 2 5 7 0.055555556 11 2 6 8 0.055555556 12 3 3 6 0.027777778 13 3 4 7 0.055555556 14 3 5 8 0.055555556 15 3 6 9 0.055555556 16 4 4 8 0.027777778 17 4 5 9 0.055555556 18 4 6 10 0.055555556 19 5 5 10 0.027777778 20 5 6 11 0.055555556 21 6 6 12 0.027777778

I add probabilities of like outcomes to get the probability that the sum of the two dice rolls are equal to k for all values of k ranging from 2 to 12.

The table below lists the name of each square and the probability of landing on each square after the first roll of two dice.

 First Roll Results in a Game of Monopoly Dice Sum Landing Spot Probability 2 Community Chest 0.027777778 3 Baltic Ave. 0.055555556 4 Income Tax 0.083333333 5 Reading Railroad 0.111111111 6 Oriental Ave 0.138888889 7 Chance 0.166666667 8 Vermont Ave 0.138888889 9 Connecticut Ave 0.111111111 10 Just Visiting 0.083333333 11 St. Charles Place 0.055555556 12 Electric Company 0.027777778 1

The square that you are most likely to land on is Chance because the most likely outcome for the sum of two dice rolls is 7.

The squares you are least likely to land on are Community Chest and Electric Company.   This occurs because there is only one way to get on Community Chest (1,1) and only one way to get on Electric Company (6,6).   All other possible sums formed by two dice rolls can be formed by more than one combination.

Note:  Readers might be interested in previous posts on the game of Risk.

More Risk and Monopoly Problems will follow.