It would be useful to compare this situation to some other situation using these techniques.
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
properties 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.
Authors Note: Go to this page for more monopoly
probability questions.
No comments:
Post a Comment