*For Exercise 4.54, see page 195; for 4.55, see page 197; for 4.56, see page 198; for 4.57 and 4.58, see pages 198–199; for 4.59 and 4.60, see page 200; and for 4.61, see page 202.*

4.62 Find and explain some probabilities.

- Can we have an event that has negative probability? Explain your answer.
- Suppose and . Explain what it means for and to be disjoint. Assuming that they are disjoint, find the probability that or occurs.
- Explain in your own words the meaning of the rule .
- Consider an event . What is the name for the event that does not occur? If , what is the probability that does not occur?
- Suppose that and are independent and that and . Explain the meaning of the event { and }, and find its probability.

4.63 Unions.

- Assume that, , and . If the events , , and are disjoint, find the probability that the union of these events occurs.
- Draw a Venn diagram to illustrate your answer to part (a).
- Find the probability of the complement of the union of , , and

4.64 Conditional probabilities.

Suppose that , , and .

- Find the probability that both and occur.
- Use a Venn diagram to explain your calculation.
- What is the probability of the event that occurs and does not?

4.65 Find the probabilities.

Suppose that the probability that occurs is 0.6 and the probability that and occur is 0.5.

- Find the probability that occurs given that occurs.
- Illustrate your calculations in part (a) using a Venn diagram.

4.66 What's wrong?

In each of the following scenarios, there is something wrong. Describe what is wrong and give a reason for your answer.

- is always equal to the sum of and .
- The probability of an event minus the probability of its complement is always equal to 1.
- Two events are disjoint if .

4.67 Attendance at two-year and four-year colleges.

In a large national population of college students, 61% attend four-year institutions and the rest attend two-year institutions. Males make up 44% of the students in the four-year institutions and 41% of the students in the two-year institutions.

207

- Find the four probabilities for each combination of gender and type of institution in the following table. Be sure that your probabilities sum to 1.
**Men****Women**Four-year institution Two-year institution - Consider randomly selecting a female student from this population. What is the probability that she attends a four-year institution?

4.68 Draw a tree diagram.

Refer to the previous exercise. Draw a tree diagram to illustrate the probabilities in a situation in which you first identify the type of institution attended and then identify the gender of the student.

4.69 Draw a different tree diagram for the same setting.

Refer to the previous two exercises. Draw a tree diagram to illustrate the probabilities in a situation in which you first identify the gender of the student and then identify the type of institution attended. Explain why the probabilities in this tree diagram are different from those that you used in the previous exercise.

4.70 Education and income.

Call a household prosperous if its income exceeds $100,000. Call the household educated if at least one of the householders completed college. Select an American household at random, and let be the event that the selected household is prosperous and the event that it is educated. According to the Current Population Survey, , , and the probability that a household is both prosperous and educated is . What is the probability that the household selected is either prosperous or educated?

4.71 Find a conditional probability.

In the setting of the previous exercise, what is the conditional probability that a household is prosperous, given that it is educated? Explain why your result shows that events and are not independent.

4.72 Draw a Venn diagram.

Draw a Venn diagram that shows the relation between the events and in Exercise 4.70. Indicate each of the following events on your diagram and use the information in Exercise 4.70 to calculate the probability of each event. Finally, describe in words what each event is.

- { and }.
- { and }.
- { and }.
- { and }.

4.73 Sales of cars and light trucks.

Motor vehicles sold to individuals are classified as either cars or light trucks (including SUVs) and as either domestic or imported. In a recent year, 69% of vehicles sold were light trucks, 78% were domestic, and 55% were domestic light trucks. Let be the event that a vehicle is a car and the event that it is imported. Write each of the following events in set notation and give its probability.

- The vehicle is a light truck.
- The vehicle is an imported car.

4.74 Conditional probabilities and independence.

Using the information in Exercise 4.73, answer these questions.

- Given that a vehicle is imported, what is the conditional probability that it is a light truck?
- Are the events “vehicle is a light truck” and “vehicle is imported” independent? Justify your answer.

4.75 Unemployment rates.

As noted in Example 4.18 (page 197), in the language of government statistics, you are “in the labor force” if you are available for work and either working or actively seeking work. The unemployment rate is the proportion of the labor force (not of the entire population) who are unemployed. Based on the table given in Example 4.18, find the unemployment rate for people with each gender. How does the unemployment rate change with gender? Explain carefully why your results suggest that gender and being employed are not independent.

4.76 Loan officer decision.

A loan officer is considering a loan request from a customer of the bank. Based on data collected from the bank's records over many years, there is an 8% chance that a customer who has overdrawn an account will default on the loan. However, there is only a 0.6% chance that a customer who has never overdrawn an account will default on the loan. Based on the customer's credit history, the loan officer believes there is a 40% chance that this customer will overdraw his account. Let be the event that the customer defaults on the loan, and let be the event that the customer overdraws his account.

- Express the three probabilities given in the problem in the notation of probability and conditional probability.
- What is the probability that the customer will default on the loan?

4.77 Loan officer decision.

Considering the information provided in the previous exercise, calculate . Show your work. Also, express this probability in words in the context of the loan officer's decision. If new information about the customer becomes available before the loan officer makes her decision, and if this information indicates that there is only a 25% chance that this customer will overdraw his account rather than a 40% chance, how does this change ?

208

4.78 High school football players.

Using the information in Example 4.21 (pages 200–201), determine the proportion of high school football players expected to play professionally in the NFL.

4.79 High school baseball players.

It is estimated that 56% of MLB players have careers of three or more years. Using the information in Example 4.22 (pages 201–202), determine the proportion of high school players expected to play three or more years in MLB.

4.80 Telemarketing.

A telemarketing company calls telephone numbers chosen at random. It finds that 70% of calls are not completed (the party does not answer or refuses to talk), that 20% result in talking to a woman, and that 10% result in talking to a man. After that point, 30% of the women and 20% of the men actually buy something. What percent of calls result in a sale? (Draw a tree diagram.)

4.81 Preparing for the GMAT.

A company that offers courses to prepare would-be MBA students for the GMAT examination finds that 40% of its customers are currently undergraduate students and 60% are college graduates. After completing the course, 50% of the undergraduates and 70% of the graduates achieve scores of at least 600 on the GMAT. Use a tree diagram to organize this information.

- What percent of customers are undergraduates
*and*score at least 600? What percent of customers are graduates*and*score at least 600? - What percent of all customers score at least 600 on the GMAT?

4.82 Sales to women.

In the setting of Exercise 4.80, what percent of sales are made to women? (Write this as a conditional probability.)

4.83 Success on the GMAT.

In the setting of Exercise 4.81, what percent of the customers who score at least 600 on the GMAT are undergraduates? (Write this as a conditional probability.)

4.84 Successful bids.

Consolidated Builders has bid on two large construction projects. The company president believes that the probability of winning the first contract (event ) is 0.6, that the probability of winning the second (event ) is 0.5, and that the probability of winning both jobs (event { and }) is 0.3. What is the probability of the event { or } that Consolidated will win at least one of the jobs?

4.85 Independence?

In the setting of the previous exercise, are events and independent? Do a calculation that proves your answer.

4.86 Successful bids, continued.

Draw a Venn diagram that illustrates the relation between events and in Exercise 4.84. Write each of the following events in terms of , , , and . Indicate the events on your diagram and use the information in Exercise 4.84 to calculate the probability of each.

- Consolidated wins both jobs.
- Consolidated wins the first job but not the second.
- Consolidated does not win the first job but does win the second.
- Consolidated does not win either job.

4.87 Credit card defaults.

The credit manager for a local department store is interested in customers who default (ultimately failed to pay entire balance). Of those customers who default, 88% were late (by a week or more) with two or more monthly payments. This prompts the manager to suggest that future credit be denied to any customer who is late with two monthly payments. Further study shows that 3% of all credit customers default on their payments and 40% of those who have not defaulted have had at least two late monthly payments in the past.

- What is the probability that a customer who has two or more late payments will default?
- Under the credit manager's policy, in a group of 100 customers who have their future credit denied, how many would we expect
*not*to default on their payments? - Does the credit manager's policy seem reasonable? Explain your response.

4.88 Examined by the IRS.

The IRS examines (audits) some tax returns in greater detail to verify that the tax reported is correct. The rates of examination vary depending on the size of the individual's adjusted gross income. In 2014, the IRS reported the percentages of total returns by adjusted gross income categories and the examination coverage (%) of returns within the given income category:^{21}

209

Income ($) |
Returns filed (%) |
Examinationcoverage (%) |
---|---|---|

None | 2.08 | 6.04 |

1 under 25K | 39.91 | 1.00 |

25K under 50K | 23.55 | 0.62 |

50K under 75K | 13.02 | 0.60 |

75K under 100K | 8.12 | 0.58 |

100K under 200K | 10.10 | 0.77 |

200K under 500K | 2.60 | 2.06 |

500K under 1MM | 0.41 | 3.79 |

1MM under 5MM | 0.19 | 9.02 |

5MM under 10MM | 0.01 | 15.98 |

10MM or more | 0.01 | 24.16 |

- Suppose a 2013 return is randomly selected and it was examined by the IRS. Use Bayes's rule to determine the probability that the individual's adjusted gross income falls in the range of $5 to $10 million. Compute the probability to at least the thousandths place.
- The IRS reports that 0.96% of all returns are examined. With the information provided, show how you can arrive at this reported percent.

4.89 Supplier Quality.

A manufacturer of an assembly product uses three different suppliers for a particular component. By means of supplier audits, the manufacturer estimates the following percentages of defective parts by supplier:

Supplier | 1 | 2 | 3 |

Percent defective | 0.4% | 0.3% | 0.6% |

Shipments from the suppliers are continually streaming to the manufacturer in small lots from each of the suppliers. As a result, the inventory of parts held by the manufacturer is a mix of parts representing the relative supplier rate from each supplier. In current inventory, there are 423 parts from Supplier 1, 367 parts from Supplier 2, and 205 parts from Supplier 3. Suppose a part is randomly chosen from inventory. Define “S1” as the event the part came from Supplier 1, “S2” as the event the part came from Supplier 2, and “S3” as the event the part came from Supplier 3. Also, define “D” as the event the part is defective.

- Based on the inventory mix, determine , , and .
- If the part is found to be defective, use Bayes's rule to determine the probability that it came from Supplier 3.