Monday, January 8, 2018

A Tough Monopoly Situation

A Tough Monopoly Situation

Situation:  The monopoly game is down to two people   Jane owns both the red and orange properties and has a hotel on each property.    The rent with one hotel is listed below.

 Jane’s Properties Saint James \$950 Tennessee \$950 New York \$1,000 Kentucky \$1,050 Indiana \$1,050 Illinois \$1,100

These are the only properti8es that Jane owns on the entire board.

Fred, the only remaining player sits on electric company.   It is his turn to throw the two die.

List all squares where Fred may land after throwing the dice.  What is the probability of each of these outcomes?

Consider the random variable R equal to the rent paid after throwing the dice.
What are the possible outcomes for random variable R?  What is the probability of each outcome?

What is the probability mass function (PMF) for the rent paid?  (The PMF of random variable R is f(R) =  P(R=k) for all k.)

What is the cumulative distribution function (CDF)  for rent paid after throwing the two die?  (The CDF for a random variable R is F(R)=P(R<=k) for all values of R.)

Analysis:

There are 36 combinations from throwing two die.   The two die must sum to a number from 2 to 11 inclusive, a total of 11 outcomes.

This was explained in a previous post.

Here are the number of die combinations for each occurrence of the die summing from 2 to 12.

 Discussion of Die Outcomes Sum  of Die Number of Combinations Probability 2 1 0.027777778 3 2 0.055555556 4 3 0.083333333 5 4 0.111111111 6 5 0.138888889 7 6 0.166666667 8 5 0.138888889 9 4 0.111111111 10 3 0.083333333 11 2 0.055555556 12 1 0.027777778 Total 36 1

The table below lists the 11 places where Fred night land and the probability of each landing when he starts at Electric Company

 Rental Outcomes When Fred Starts from Electric Company Sum of Die Landing Probability Probability Rent 2 Virginia Ave 0.027777778 0 3 Pennsylvania Railroad 0.055555556 0 4 St. James Place 0.083333333 950 5 Community Chest 0.111111111 0 6 Tennessee Avenue 0.138888889 950 7 New York Avenue 0.166666667 1000 8 Free Parking 0.138888889 0 9 Kentucky Avenue 0.111111111 1050 10 Chance 0.083333333 0 11 Indiana Avenue 0.055555556 1050 12 Illinois Avenue 0.027777778 1100

The PMF of the rent random variable is obtained by finding the sum pf the probabilities for each rent=k.   Note there are five possible rent outcomes 0, 950, 1000, 1050, and 1100.

The probability that the rent is 0 is the sum of the probabilities for the five dice sums that lead to Fred not landing on a property owned by Jana.

The same process is used to find the probability rent is 950, 1000, 1050 or 1100.

The CDF of the rent random variable is obtained by summing the probability of all events where rent  <K.

When k=-0 the CDF is P(rent=0)

When K=950 the CDF is P(rent=0) + P(rent=950)

An so on.

The PMF and the CDF of the rental outcomes are below.

 PMF and CDF for the Rent Random Variable Rent pmf cdf 0 0.41667 0.41667 950 0.22222 0.63889 1000 0.16667 0.80556 1050 0.16667 0.97222 1100 0.02778 1.00000 1.0000

Authors Note: The actual rules of monopoly require that a person who throws doubles go again prior to losing his term.   Also, a person that throws doubles three times goes to jail and I believe does not pay rent on the third throw.

Alternative Question:  What is the PMF and the CDF of Fred’s rent under the conditions of this monopoly board if he completes his full turn?

Authors Note:   If you enjoyed this post you probably will find a book that I wrote in 1997 called Statistical Applications of Baseball highly useful.   Go here to purchase the book on Kindle.