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Mathematics homework help

Milestone 2 Worksheet

In this milestone, you will explore probability. Even in simple situations, it’s easy for one particular outcome to be highly unlikely. We’ll learn how even a simple choice yields many possibilities.

 

Part 1: Choose a Paint Color for your House

 

First list below the type and number of rooms contained in your dream home.

List of rooms:

Total Rooms:

 

Scenario 1: Imagine that you can paint your room either beige or grey.

How many different ways can you paint each room in your house either beige or grey? __________

 

Scenario 2: Of course, you’d like to have more than two choices of paint! Let’s first find the total number of options we have to choose from!

 

Total Shades of Red Available: 317

Total Shades of Orange Available: 521

Total Shades of Yellow Available: 516

Total Shades of Green Available: 626

Total Shades of Blue Available: 441

Total Shades of Purple Available: 496

Total Shades of Gray Available: 191

Total Shades of Beige Available: 312

TOTAL Possible Shades of Colors Available: ________

Determine how many possible ways you could paint your house, given this color scheme, assuming that each room is painted one distinct color and rooms do not need to be different colors. Explain your answer below:

 

 

 

 

 

 

 

 

 

 

 

Part 2: Applying for Credit Cards

 

When you open credit cards at a bank, you initiate a hard inquiry once you apply for credit. If you are approved for a credit card, this counts as a new account. Having too many hard inquiries and new accounts on your credit report lowers your credit score, so you’ll want to be judicious when you apply for additional lines of credit.

 

Let’s say you’re interested in the following credit cards:

 

 

  • Chase Sapphire Reserve
  • Wells Fargo Propel Card
  • American Express Blue Cash Everyday
  • Citi Double Cash Card
  • Chase Freedom
  • Chase Amazon Prime Rewards Card
  • Bank of America Cash Rewards Visa
  • American Express Platinum

 

 

Answer the following questions below (show work as applicable, using factorial notation if necessary):

 

Scenario 1: First imagine that you would like to apply for all of these credit cards right now. How many different ways could you apply for these cards? ____________

 

Scenario 2: Imagine you would like to apply for three of these credit cards. How many different ways could you apply for three credit cards out of this group, without taking order into account? _____________

 

Scenario 3: This one is going to be a little more challenging. Some companies, such as Chase, have strict underwriting guidelines. Chase will automatically reject any credit card application if you’ve opened at least five accounts in the past twenty-four months. Let’s explore this scenario a bit further:

 

Part A: Once again, imagine that you want to apply for three credit cards out of the eight above. However, you now want to take order into account. In Scenario 2, we did not care what order we applied for the group of three credit cards… in this case, order does matter, because we may get declined depending on the order we apply for our new credit cards.

 

Explain how to find the total number of ways to apply for these three cards, with respect to order, below:

 

 

 

 

 

 

 

 

 

 

 

Part B: Let’s imagine that we want to apply for Chase cards first, so that we aren’t declined by the 5/24 rule stated above. So your top three picks must be: Chase Freedom, Chase Sapphire Reserve, and Chase Amazon Prime (not necessarily in that order!). Then you can apply for the final five cards in any order you wish.

 

Explain how to find the total number of ways you could apply for these cards, given the constraints above:

 

 

 

 

 

 

 

 

 

 

 

 

Part 3: Insurance

 

As a homeowner, you are required to purchase insurance by your lender. This is a non-negotiable expense, as this policy protects you and the lender from damages to the property. For most people, insurance serves as a safeguard against unexpected losses. But how do insurance companies make money? The people responsible for helping these companies make money are known as actuaries, who use probability theory to determine the expectation of loss and profit.

 

Part A: Determining the Odds of Damages.

 

An actuary is looking into the damages caused by water damage caused by freezing (F) or flooding (FL) within her assigned region. She uses historical data to calculate the probability of damages as follows:

.

 

How would the actuary calculate the probability that water damage is caused by frozen pipes or flooding? Show your work below:

 

 

 

 

 

 

 

 

 

 

 

Next the actuary wants to investigate theft. She has found overall that those who purchase security systems encounter a lower rate of theft. Approximately 35% of people in the neighborhood have an active security system in place. She wants to narrow the marketing of this insurance plan to only people who have active security systems. She wants to find the conditional probability that these homeowners encounter theft, given that they own a security system.

 

She found historically that 0.2% of the total population in her region own a security system and encountered theft. How can she find the probability that someone encounters theft, given that they own a security system? Round your answer to four decimal places. Explain your answer below:

 

 

 

 

 

 

 

 

 

 

 

Part B: Calculating the expected value of damages.

 

Now our actuary has found the probability of each possible event happening over the year. She has also calculated the average damages associated with each claim in the table below. Fill in the probability of damages from Part 3A in the table!

 

Type of Claim Probability of Damages Cost of Damages
Fire and Lightning 0.35% $43,983
Wind and Hail 2.86% $8,313
Flooding and Freezing _____ $8,861
Theft _____ $3,990
House Fire 0.54% $54,208
Other Damages 0.88% $5,048

 

For her market, she would like to charge approximately $1150 per year for her. She wants to find the average rate of return for each of her customers. First she needs to find the expected value of damages—that is, how much the insurance provider expects to pay for each person. Use the table above to find the expected value of damages and explain your answer in the box below:

 

 

 

 

 

 

 

Let’s assume that the target market for this insurance policy is approximately 22,500 people. If the insurance provider were able to sell this policy to 40% of her target market at $1150 per year, how much money would the company make with this policy? Show all of your work below:

 

 

 

 

 

 

 

 

 

 

 

 

Part 4: Reflection

 

Please answer each of the following questions with at least 2-3 complete paragraphs per question below:

 

Question 1: Give an example where it is easy to create many possibilities within your career. Given your studies on counting, how might even a short list of possible decisions yield many possibilities? Why might it be a better idea to narrow down your possible choices?

 

 

 

 

 

 

Question 2: How is a permutation different from a combination? Provide a few (2-3) examples of when a combination would be more appropriate than a permutation to determine the number of possibilities. Explain why your examples require the use of combinations.

 

 

 

 

 

 

 

Question 3: How do insurance and warranty providers make money? What kinds of products should you ensure and what kinds of products should you not insure?

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