Saturday, August 17, 2019

Some Issues With Electric Company

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?

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