## Friday, May 13, 2016

### Monopoly PR6: Rent on Yellow Properties

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 non-yellow 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 non-yellows are 1-14/36, 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(Rent-E(Rent)2)1/2

The formula for the skew is E(Rent-E(Rent))3/SD3

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 (rent-E(rent)2 (Rent-E(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 (rent-E(rent)2 (Rent-E(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.