3.12 Assignment 3

Purposes

The following questions assess your knowledge of identifying the sample space of a chance experiment, calculating probabilities using equally likely outcome model (the f/N rule) if applicable, calculating probabilities of events using addition, complementation, conditional, multiplication rules, showing whether two events are independent by calculation, and using combination and permutation to calculate the number of sample points of events.

Resources

M03_SaleHome_Recode.xlsx

Instructions

Complete the following:

1. Let H be the event of observing a head and T be the event of observing a tail. A balanced coin is tossed three times.
   1. List all possible outcomes. (2 marks)
   2. List all possible outcomes and find the probabilities of the following events. (8 marks)
      1. A = event exactly two heads are tossed
      2. B = event the first toss is a tail
      3. C = event the first toss is a head
      4. D = event all three tosses come up the same
   3. List all possible outcomes and find the probabilities of the following events. (10 marks: 2 for each)
      1. not A
      2. A & B
      3. B & C
      4. C or D
      5. D|A
   4. Identify all possible pairs of events defined in part (b) that are mutually exclusive. (3 marks)
   5. Are the events A and D independent? Explain your answer mathematically. (3 marks)
2. A contingency table for injuries in the United States by circumstance (column variable) and gender (row variable) is given as follows. Note that frequencies are in millions.

1. 1. Complete the contingency table. (3 marks)
   2. Find the probability that an injured person was hurt at work, that is, [latex]P(C_{1})[/latex]. (2 marks)
   3. Find the probability that an injured person was hurt at work and she was a female, that is, [latex]P(C_{1}\ \&\ R_{2})[/latex]. (2 marks)
   4. Find the probability that a female injured person was hurt at work, that is, [latex]P(C_{1}|R_{2})[/latex]. (3 marks)
   5. Are events [latex]C_{1}[/latex] and [latex]R_{2}[/latex] independent? Explain your answer. (2 marks)
   6. Are events [latex]C_{1}[/latex] and [latex]R_{2}[/latex] mutually exclusive? Explain your answer. (2 marks)
   7. Is the event that an injured person is male independent of the event that an injured person was hurt at home? Explain by calculation. (4 marks)
2. Consider a population consisting of 60 students.
   1. How many samples of size 5 are possible? (2 marks)
   2. What is the probability of taking each sample? (2 marks)
3. The U.S. Senate consists of 100 senators, two from each state. A committee consisting of five senators is to be formed.
   1. How many different committees are possible? (3 marks)
   2. How many different committees are possible if no state may have more than one senator on the committee? (4 marks)
4. Suppose that a rare disease occurs in the general population in only one of every 10,000 people. A medical test is used to detect the disease. If a person has the disease, the probability that the test result is positive is 0.99. If a person does not have the disease, the probability that the test result is positive is 0.02. Given that a person’s test result is positive, find the probability that this person truly has the rare disease. (Bonus: 5 marks)

Part B

Finish the following questions using R and R commander:

Read the data set “M03_SaleHome_Recode.xlsx” and use R commander to complete the following tasks. For each, you need to copy or do a screenshot of the output in R commander (we later call it computer output) and paste it into the space below the questions. To save space, you only need to copy and paste what is asked for in the questions and you can shrink the size of the image if necessary.

Use R commander to obtain a proper frequency distribution or contingency table to answer the following questions.

1. If we randomly select a sale home, what is the probability that it is a small house? (4 marks)
2. If we randomly select a sale home, what is the probability that it is a small house with a swimming pool? (4 marks)
3. If we randomly select a sale home, what is the probability that it is a small house given that it has a swimming pool? (4 marks)
4. If we randomly select a sale home that has a tiled roof, what is the probability that it is a small house with a swimming pool? (4 marks)
5. If we randomly select a small house, what is the probability that this house has a tiled roof given that it does not have a swimming pool? (4 marks)