A title company wants more information about mortgages based on the type of home being bought. 1000 mortgages are randomly selected. The sample results in the table below shows the number of mortgages funded based on the type of home purchased. Answer the following questions. Include your answers in fraction and in decimal forms and round your decimal answers to 4 digits.

Question

A title company wants more information about mortgages based on the type of home being bought.  1000 mortgages are randomly selected.  The sample results in the table below shows the number of mortgages funded based on the type of home purchased.  Answer the following questions. Include your answers in fraction and in decimal forms and round your decimal answers to 4 digits. 
	Single-Family	Condo	Multifamily	Total
Adjustable	400		x	x                   [?]
Fixed	[?]	90		[?]      300   
Total		[?]  [?]	200	1000
Fill in the remaining cells in the above table with your own numbers ensuring rows sum to row totals and columns sum to column totals.  In addition, the row and column totals will both sum to the grand total of 1000.  Answers will and should vary.  The following questions will use the numbers (already existing or added by you) in this table. Fill in the table cells very carefully and use this table to answer the following questions.

a.	What is the probability that the mortgage is Adjustable AND for a Single Family? 
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b.	What is the probability that the type of home purchased is a Condo?
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c.	What is the probability that the mortgage is Fixed OR a Single-Family home is purchased? 
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d.	Given that a Condo is purchased, what is the probability the mortgage is Adjustable?
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e.	Given that the mortgage is Adjustable, what is the probability that a Condo is purchased?
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f.	Determine if type of mortgage and home purchased are independent. Justify your answer by showing your work here. (HINT: Use the Multiplication Rule or conditional probabilities from above)

g.	Determine if purchasing a Single-Family home and a Condo are mutually exclusive events. Justify your answer using probabilities.

GPT-4o mini · Answer

Here is a completed version of the given table with numbers selected and ensuring that rows and columns sum to the specified totals.

Updated Table

| | Single-Family | Condo | Multifamily | Total | |-------------------|----------------|-------|-------------|-------| | Adjustable | 400 | 100 | 100 | 600 | | Fixed | 200 | 90 | 110 | 400 | | Total | 600 | 190 | 210 | 1000 |

Now let's answer the questions based on this table:

a. What is the probability that the mortgage is Adjustable AND for a Single Family?

This is found by dividing the number of Adjustable Single-Family mortgages by the total number of mortgages:

\[ P(\text{Adjustable} \cap \text{Single-Family}) = \frac{400}{1000} = \frac{4}{10} = 0.4000 \]

b. What is the probability that the type of home purchased is a Condo?

This is found by dividing the number of Condos by the total number of mortgages:

\[ P(\text{Condo}) = \frac{190}{1000} = \frac{19}{100} = 0.1900 \]

c. What is the probability that the mortgage is Fixed OR a Single-Family home is purchased?

To find this we can use the formula:

\[ P(\text{Fixed} \cup \text{Single-Family}) = P(\text{Fixed}) + P(\text{Single-Family}) - P(\text{Fixed} \cap \text{Single-Family}) \]

Calculating each probability:

$ P(\text{Fixed}) = \frac{400}{1000} = 0.4000 $
$ P(\text{Single-Family}) = \frac{600}{1000} = 0.6000 $
$ P(\text{Fixed} \cap \text{Single-Family}) = \frac{200}{1000} = 0.2000 $

So,

\[ P(\text{Fixed} \cup \text{Single-Family}) = 0.4000 + 0.6000 - 0.2000 = 0.8000 \]

d. Given that a Condo is purchased, what is the probability the mortgage is Adjustable?

This is found using conditional probability:

\[ P(\text{Adjustable} | \text{Condo}) = \frac{P(\text{Adjustable} \cap \text{Condo})}{P(\text{Condo})} = \frac{100/1000}{190/1000} = \frac{100}{190} \approx 0.5263 \]

e. Given that the mortgage is Adjustable, what is the probability that a Condo is purchased?

This is similarly found:

\[ P(\text{Condo} | \text{Adjustable}) = \frac{P(\text{Condo} \cap \text{Adjustable})}{P(\text{Adjustable})} = \frac{100/1000}{600/1000} = \frac{100}{600} \approx 0.1667 \]

f. Determine if the type of mortgage and home purchased are independent.

Two events A (Mortgage type) and B (Home type) are independent if $ P(A \cap B) = P(A) \cdot P(B) $.

Let's check if adjustable mortgages and Single Family homes are independent:

$ P(\text{Adjustable}) = \frac{600}{1000} = 0.6000 $
$ P(\text{Single-Family}) = \frac{600}{1000} = 0.6000 $
$ P(\text{Adjustable} \cap \text{Single-Family}) = \frac{400}{1000} = 0.4000 $

Now, $ P(\text{Adjustable}) \cdot P(\text{Single-Family}) = 0.6000 \cdot 0.6000 = 0.3600 $.

Since $ 0.4000 \neq 0.3600 $, the events are not independent.

g. Determine if purchasing a Single-Family home and a Condo are mutually exclusive events.

Two events A and B are mutually exclusive if they cannot both occur at the same time $ P(A \cap B) = 0 $.

In this case, a mortgage can be both Single-Family and Condo. Thus,

\[ P(\text{Single-Family} \cap \text{Condo}) \neq 0 \]

Hence, they are not mutually exclusive events.

This concludes the answers needed based on the given table and assumptions made.

Answer

The table should read Adjustable- Single Family- 400, condo- 100, multifamilty- 200, total - 700 Fixed- single family- 210, condo- 90, multifamily- 0, total - 300 now give me your answers for A-G

GPT-4o mini · Answer

Based on the updated table provided: ### Updated Table | | Single-Family | Condo | Multifamily | Total | |-------------------|----------------|-------|-------------|-------| | Adjustable | 400 | 100 | 200 | 700 | | Fixed | 210 | 90 | 0 | 300 | | Total | 610 | 190 | 200 | 1000 | Now, let's compute the probabilities for questions A-G based on this table. ### Answers **a. What is the probability that the mortgage is Adjustable AND for a Single Family?** $$ P(\text{Adjustable} \cap \text{Single-Family}) = \frac{400}{1000} = \frac{4}{10} = 0.4000 $$ **b. What is the probability that the type of home purchased is a Condo?** $$ P(\text{Condo}) = \frac{190}{1000} = \frac{19}{100} = 0.1900 $$ **c. What is the probability that the mortgage is Fixed OR a Single-Family home is purchased?** We can use the formula: $$ P(\text{Fixed} \cup \text{Single-Family}) = P(\text{Fixed}) + P(\text{Single-Family}) - P(\text{Fixed} \cap \text{Single-Family}) $$ Calculating each probability: - $ P(\text{Fixed}) = \frac{300}{1000} = 0.3000 $ - $ P(\text{Single-Family}) = \frac{610}{1000} = 0.6100 $ - $ P(\text{Fixed} \cap \text{Single-Family}) = \frac{210}{1000} = 0.2100 $ So, $$ P(\text{Fixed} \cup \text{Single-Family}) = 0.3000 + 0.6100 - 0.2100 = 0.7000 $$ **d. Given that a Condo is purchased, what is the probability the mortgage is Adjustable?** Using conditional probability: $$ P(\text{Adjustable} | \text{Condo}) = \frac{P(\text{Adjustable} \cap \text{Condo})}{P(\text{Condo})} = \frac{100/1000}{190/1000} = \frac{100}{190} \approx 0.5263 $$ **e. Given that the mortgage is Adjustable, what is the probability that a Condo is purchased?** Using conditional probability: $$ P(\text{Condo} | \text{Adjustable}) = \frac{P(\text{Condo} \cap \text{Adjustable})}{P(\text{Adjustable})} = \frac{100/1000}{700/1000} = \frac{100}{700} \approx 0.1429 $$ **f. Determine if the type of mortgage and home purchased are independent.** To check independence, we check if $ P(A \cap B) = P(A) \cdot P(B) $. Assuming the events A = Adjustable and B = Single-Family: - $ P(\text{Adjustable}) = \frac{700}{1000} = 0.7000 $ - $ P(\text{Single-Family}) = \frac{610}{1000} = 0.6100 $ - $ P(\text{Adjustable} \cap \text{Single-Family}) = \frac{400}{1000} = 0.4000 $ Now, calculate $ P(\text{Adjustable}) \cdot P(\text{Single-Family}) $: $$ P(\text{Adjustable}) \cdot P(\text{Single-Family}) = 0.7000 \cdot 0.6100 = 0.4270 $$ Since $ P(\text{Adjustable} \cap \text{Single-Family}) = 0.4000 \neq 0.4270 $, the events are **not independent**. **g. Determine if purchasing a Single-Family home and a Condo are mutually exclusive events.** Two events are mutually exclusive if $ P(A \cap B) = 0 $. Here, we can have both a Single-Family and a Condo mortgage, therefore, $$ P(\text{Single-Family} \cap \text{Condo}) \neq 0 $$ So, they are **not mutually exclusive events**. ### Summary of Answers: - **a:** $ \frac{4}{10} $ or $ 0.4000 $ - **b:** $ \frac{19}{100} $ or $ 0.1900 $ - **c:** $ 0.7000 $ - **d:** $ \approx 0.5263 $ - **e:** $ \approx 0.1429 $ - **f:** Not independent (based on calculations) - **g:** Not mutually exclusive (based on definitions)