In this problem the most likely amount owed and the expected amount owed are the same. Often these values differ as shown in the second half of the problem.
Some Issues With Electric Company
Question: Sally owns electric company but does not own
waterworks. Mark lands on electric
company. The rent on electric company
is 4 times the sum of two dice throws. What is the value of each possible
amount that Mark will owe Sally and the probability of that outcome? What is most likely amount that Mark will owe
Sally? What is the expected amount that
Mark will owe Sally?
Sally
buys waterworks and Mark lands on electric company again. The rent is now 10 times the sum of the two
dice rolls. What is the most likely
amount that Mark will owe Sally? What
is the expected amount?
In these
two problems the most likely amount owed and the expected amount owed are
equal. Is this always the case? Why or why not?
What is
the most likely payoff from landing on electric company and the expected rent from landing on electric company if the rent is the
square of the sum of the two dice rolls in dollars?
Note: The probabilities for the sum of two dice
throws were calculated in a previous post.
Analysis:
The
electric company rent calculations for the situation where the owner has only
electric company and the situation where the owner has both utilities is
presented below.
Rent From Landing on
Electric Company


Dice Sum

Rent without waterworks

Rent with waterworks

Probability

2

8

20

0.027777778

3

12

30

0.055555556

4

16

40

0.083333333

5

20

50

0.111111111

6

24

60

0.138888889

7

28

70

0.166666667

8

32

80

0.138888889

9

36

90

0.111111111

10

40

100

0.083333333

11

44

110

0.055555556

12

48

120

0.027777778

The
expected rent is the SUMPRODUCT of the rent level and the probability of the
dice roll outcome.
Expected Rent from
Landing On Electric Company


Electric Company Only

28

Electric Company and
Waterworks

70

The most
likely dice roll outcome in this case is 7.
The rents in this case are 28 for the person who owns one utility and 70
for the person who owns both utilities.
In this problem the expected outcome is equal to the most likely
outcome.
Is the
expected outcome always equal to the most likely outcome? THE ANSWER IS AN EMPHATIC NO!!!!!!!
What if
rents were determined by a coin toss?
In this situation, the only two outcomes are equally likely and the
expected outcome (the weighted average of the two outcomes) could never occur.
The
expected outcome differs from the most likely outcome when rent is equal to the
sum of the dice roll squared.
This
calculation is presented below.
When Rents are Equal to
the Sum
of the Dice Squared


Dice Sum

Rent equal to Dice Sum
Squared

Probability

2

4

0.027777778

3

9

0.055555556

4

16

0.083333333

5

25

0.111111111

6

36

0.138888889

7

49

0.166666667

8

64

0.138888889

9

81

0.111111111

10

100

0.083333333

11

121

0.055555556

12

144

0.027777778

Expected Rent

54.8


Most likely Rent

49

Authors Note: A previous post considers the impact of
throwing doubles in monopoly. The
player who throws doubles gets to go again.
The player who throws doubles twice goes to jail. One can no longer define an outcome of the experiment
exclusively by the square a person lands on because it is possible for a person
to land on more than one square.
Go to the
following post to consider this problem.
Throwing
Doubles:
Using the
technique presented in this post and the throwing doubles post consider the
following problem.
Extra Problem: Mary is seven squares away from electric
company. Fred owns both Electric Company
and Waterworks. What is the expected
rent and the most likely rent that Mary will have to pay rent after her next
monopoly turn?
The
reader of this post might benefit from these online resources.
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