Question: A person starts on Kentucky Avenue and
throws the dice once. (We will consider
ramifications of doubles in a future post but for now let’s assume the person
does NOT go again if he throws doubles.)
The player’s opponent owns all three yellow properties
 Atlantic, Ventnor, and Marvin
Gardens. The rents at these three properties for two
houses and for three houses are presented below.
Rent at Atlantic, Ventor,
and Marvin Gardens


Two Houses

Three Houses


Atlantic

330

800

Ventor

330

800

Marvin Gardens

360

850

What is the likelihood that the person will not land on any
of these three properties after one throw of the two dice?
What is the most likely outcome after one throw of the two
dice?
What are the expected value, standard deviation, and
skewness of rents given two house and given three houses?
Analysis: The landing probabilities for one throws of
the two dice from the starting point of Go were calculated in a previous post.
The table below
updates these landing probabilities for the situation where the starting point
is Kentucky Avenue.
Outcomes from One Roll of
Two Dice Starting From Kentucky


Sum of Dice Rolls

Property Landed On

Probability

2

Indiana

0.027777778

3

Illinois

0.055555556

4

B&O Railroad

0.083333333

5

Atlantic Ave

0.111111111

6

Ventnor Ave

0.138888889

7

Water Works

0.166666667

8

Marvin Garden

0.138888889

9

Go to Jail

0.111111111

10

Pacific Ave

0.083333333

11

North Carolina Ave

0.055555556

12

Community Chest

0.027777778

The probability of not landing on any of the three yellow
properties is 0.611111. You can get
this by summing up probabilities for all nonyellow properties or you can sum
the probabilities for the yellows and subtract from one. Remember the probability of the complement
of an event is 1 probability of the event.
The probability of landing on Atlantic from Kentucky is
4/36. The probabilities for Ventnor and
Marvin Gardens are 5/36. The
probabilities for all nonyellows are 114/36, which is 0.611111.
The square the person is most likely to land on is Water
Works because the probability of getting a 7 on the sum of the two dice is
higher than any other probability.
The table below outlines calculations for expected value
Outcomes from One Roll of Two Dice Starting From Kentucky


Sum of Dice Rolls

Property Landed On

Probability

Rent Two Houses

Rent Three Houses

2

Indiana

0.027777778

0

0

3

Illinois

0.055555556

0

0

4

B&O
Railroad

0.083333333

0

0

5

Atlantic Ave

0.111111111

330

800

6

Ventor
Ave

0.138888889

330

800

7

Water Works

0.166666667

0

0

8

Marvin
Garden

0.138888889

380

850

9

Go to Jail

0.111111111

0

0

10

Pacific
Ave

0.083333333

0

0

11

North Carolina Ave

0.055555556

0

0

12

Community
Chest

0.027777778

0

0

Expected
Rent Two Houses

135.28


Expected Rent Three Houses

318.06

The formula for the standard deviation is SD= (E(RentE(Rent)^{2})^{1/2}
The formula for the skew is E(RentE(Rent))^{3/}SD^{3}
Results for the case with two houses are presented below.
Expected Rent, STD Rent,
Skewness Rent With Two Houses


Property Landed On

Probability

Rent Two Houses

Expected Rent

(rentE(rent)2

(RentE(rent))3

Indiana

0.028

0

135.3

18300

2475594

Illinois

0.056

0

135.3

18300

2475594

B&O Railroad

0.083

0

135.3

18300

2475594

Atlantic Ave

0.111

330

135.3

37917

7383233

Ventor Ave

0.139

330

135.3

37917

7383233

Water Works

0.167

0

135.3

18300

2475594

Marvin Garden

0.139

380

135.3

59889

14656161

Go to Jail

0.111

0

135.3

18300

2475594

Pacific Ave

0.083

0

135.3

18300

2475594

North Carolina Ave

0.056

0

135.3

18300

2475594

Community Chest

0.028

0

135.3

18300

2475594

Expected Rent

135.3


STD (Rent)

170.2


Skewness (Rent)

0.5

Switch the rent numbers in the spreadsheet and get the
results for three houses.
Results with Thee Houses


Property Landed On

Probability

Rent Two Houses

Expected Rent

(rentE(rent)2

(RentE(rent))3

Indiana

0.028

0

135.3

18300

2475594

Illinois

0.056

0

135.3

18300

2475594

B&O Railroad

0.083

0

135.3

18300

2475594

Atlantic Ave

0.111

800

135.3

441856

293711258

Ventor Ave

0.139

800

135.3

441856

293711258

Water Works

0.167

0

135.3

18300

2475594

Marvin Garden

0.139

850

135.3

510828

365100020

Go to Jail

0.111

0

135.3

18300

2475594

Pacific Ave

0.083

0

135.3

18300

2475594

North Carolina Ave

0.056

0

135.3

18300

2475594

Community Chest

0.028

0

135.3

18300

2475594

Expected Rent

318.1


STD (Rent)

438.9


Skewness (Rent)

1.5

The increase in rent due to the addition of the third house
increases expected rent, variability of rent and the skewness. The increase in skewness occurs because the
financial differential of falling or not falling on the yellow properties
increase.
In life you might want to even out cash flows by spreading
houses on more property. In monopoly
you want to bankrupt the opponent, which is often done by concentrating your
buildings on a narrower set of properties.
Additional Work: The monopoly problems can be extended in a
number of directions. Often in the game
a person has two monopolies and is faced with a choice of concentrating
building on one monopoly or spreading building on many monopolies. This issue can be studied.
Also, in previous posts I looked at the probability of
landing on different squares after throwing doubles and going a second time.
This post looks at the probability of moving seven squares
including outcomes when the player throws a double on the first throw.
The post looks at the probability of going 15 squares on one
turn. (Hint: this requires the player
throw doubles on the first turn.)
The current problem of landing on yellows can be modified to
consider additional paths to the yellows once the player throws a double.
Next week I hope to create a post with 5 or 10 monopoly math
problems with links to the answers.
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