RMI 2101 Spring 2016 Homework Assignment 4

September 11, 2016

Question

Homework #

Due date

RMI 2101
Spring 2016
Homework Assignment 4
24 points

Due on Thursday, February 18, 2016 AT THE BEGINNING OF YOUR CLASS

Be careful to follow the “Guidelines for Homework” document located in the Course Documents Section of Blackboard. I will deduct 1 point from your grade for not properly following the homework guidelines.

Since this homework is quantitative, you should be very careful to show your work.All work must be typed.

1. Presidential primary season is upon us! Donald Trump appears to be the Republican favorite for the upcoming South Carolina primary on February 20. As of February 11, the website FiveThirtyEight projects that Trump has a 55% chance of winning the South Carolina primary (.fivethirtyeight.com/election-2016/primary-forecast/south-carolina-republican/”>link). They also project the chances of other candidates winning, including Marco Rubio (22%), Ted Cruz (15%), Jeb Bush (6%), and Ben Carson (2%). Assume that ties are not possible and that voters can only select from the above candidates.

a. What is the random variable associated with this prediction? [1 point]

b. Is this a mutually exclusive event? Why or why not? [1 point]

c. Provide a well-labeled probability distribution table based on allpredicted outcomes of thisprimary. [3 points]

d. Your friend Jean is a big Marco Rubio fan. She thinks Trump doesn’t have a chance. She proposes a bet, where she will pay you \$100 if Trump wins and you will pay her \$50 if Rubio wins. The bet is invalid if any other candidate wins. What is the expected value of this bet from your perspective? [2 points]

2. There are six Republican primaries in the month of March. Those primaries (and projected probabilities of Trump winning) are listed in the following table. Assume that each primary is independent of the others.

Date

State

Trump Win Probability

March 1

Georgia

44.0%

March 1

Texas

27.1%

March 8

Michigan

60.0%

March 15

Florida

54.0%

March 15

North Carolina

43.0%

March 22

Arizona

38.1%

a. What is the probability that Trump wins ALL the primaries in March? [3 points]

b. Consider the primaries where Trump has less than a 50% projected win probability. What is the probability that TrumpLOSES ONLY ONE of those primaries?Be sure to show your work. (Hint: he could lose Georgia AND win the others OR he could lose Texas AND win the others…) [4 points]

c. You were instructed to assume that each primary was independent of the others. This is a very strong (and probably incorrect) assumption. Why would this be an incorrect assumption? [2 points]

3. The Civil Air Patrol (CAP) owns airplanes used for search and rescue missions. In the course of their activities, CAP sustains losses in the form of property damages to their airplanes. Assume that the CAP covers two major regions in Pennsylvania– the Northeast (NE) and the Southwest (SW).

The CAP has the following data for the number of accidents per airplane in the NE Region.

# of accidents per airplane per year

Probability of having this # of accidents

0

0.60

1

0.25

2

0.10

3

0.05

a. CAP has calculated the variance for the NE Region frequency distribution to be equal to 0.74. Since you want to be sure you are using correct numbers in your evaluation, prove that CAP calculated the correct variance for frequency for the NE Region. Show all work! Round each calculation to 4 decimal places. You must use this validated variance (0.74) for all further calculations. What are the units of measurement? [4 points]

b. You have calculated the following information related to the number of accidents per airplane for the SW Region. You have had someone else check your calculations and are sure that these numbers are valid. The SW Region has 1000 planes.

Mean = 1.10

Variance = 0.81

Suppose that each region has a separate property insurance contract and that you are going to negotiate with the insurance company about insurance premiums for the NE and SW Regions.

Which region faces the most risk and why? Show all calculations and explain your numerical results. [2 points]

c. Suppose the insurance company can also cover the NE and SW Regions under a single insurance contract rather than two separate policies. The total premium under the combined policy is less than the total cost (sum of the two premiums) for insuring each of the regions separately. Provide a careful explanation as to why this might be the case. [2 points]

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