Homework Assignment 2 Due date: 10/6

September 29, 2021

1. (20 pts) Each of a sample of four home mortgages is classified as fixed rate (F ) or variable rate (V ).

a) (5 pts) What is the sample space S? List all the outcomes. b) (5 pts) Which outcomes are in the event that all four mortgages are of the

same type?

c) (5 pts) Which outcomes are in the event that at most one of the four is a variable-rate mortgage?

d) (5 pts) What is the union of the events in parts b) and c), and what is the intersection of these two events?

2. (15 pts) A certain system can experience two different types of defects. Let Ai(i = 1, 2) denote the event that the system has a defect of type i. Suppose that P (A1) = 0.12, P (A2) = 0.07 and P (A1 ?A2) = 0.13. Express the following events in terms of A1 and A2, and find the desired probabilities.

a) (5 pts) What is the probability that the system does not have a type 1 defect?

b) (5 pts) What is the probability that the system has both type 1 and type 2 defects?

c) (5 pts) What is the probability that the system has a type 1 defect but not a type 2 defect?

3. (15 pts) Computer keyboard failures can be attributed to electrical defects or mechanical defects. A repair facility currently has 15 failed keyboards, 6 of which have electrical defects and 9 of which have mechanical defects.

a) (5 pts) How many ways are there to randomly select 5 of these key boards for a thorough inspection (without regard to order)?

b) (5 pts) In how many ways can a sample of 5 keyboards be selected so that exactly two have a mechanical defect?

1

c) (5 pts) If a sample of 5 keyboards is randomly selected, what is the probability that at least 4 of these will have an electrical defect?

4. (15 pts) Consider the following information about travelers on vacation (based partly on a recent Travelocity poll): 40% check work email, 30% use a cell phone to stay connected to work, 25% bring a laptop with them, 23% both check work email and use a cell phone to stay connected, and 51% neither check work email nor use a cell phone to stay connected nor bring a laptop. In addition, 88 out of every 100 who bring a laptop also check work email, and 70 out of every 100 who use a cell phone to stay connected also bring a laptop. Hint: Let E, C and L be the events “checking work email,” “using a cell phone,” and “bringing a laptop,” respectively. We have

P (E) = 0.4, P (C) = 0.3, P (L) = 0.25, P (E ?C) = 0.23,

P (E? ?C? ?L?) = 0.51, P (E|L) = 0.88, P (L|C) = 0.7.

Then, P (E ?L) = P (E|L)P (L) = 0.22

P (C ?L) = P (L|C)P (C) = 0.21

P (E ?C ?L) = 1 ?P (E? ?C? ?L?) = 0.49

a) (5 pts) What is the probability that a randomly selected traveler who checks work email also uses a cell phone to stay connected?

b) (5 pts) What is the probability that someone who brings a laptop on vacation also uses a cell phone to stay connected?

c) (5 pts) If the randomly selected traveler checked work email and brought a laptop, what is the probability that he/she uses a cell phone to stay con- nected?

5. (10 pts) There has been a great deal of controversy over the last several years regarding what types of surveillance are appropriate to prevent terrorism. Sup- pose a particular surveillance system has a 99% chance of correctly identifying a future terrorist and a 99.9% chance of correctly identifying someone who is not a future terrorist. If there are 1000 future terrorists in a population of 300 million, and one of these 300 million is randomly selected, scrutinized by the system, and identified as a future terrorist, what is the probability that he/she actually is a future terrorist?

6. (15 pts) Components arriving at a distributor are checked for defects by two differ- ent inspectors (each component is checked by both inspectors). The first inspector detects 90% of all defectives that are present, and the second inspector does like- wise. At least one inspector does not detect a defect on 15% of all defective components.

a) (5 pts) Do these two inspectors work independently of each other?

2

b) (5 pts) What is the probability that a defective component will be detected only by the first inspector?

c) (5 pts) What is the probability that a defective component escapes detection by both inspectors?

7. (10 pts) Suppose that a certain company sends 40% of its overnight mail parcels via express mail service E1, and sends the remaining 60% via E2. Of these parcels sent via E1, 5% arrive late, whereas only 1% of those handled by E2 arrive late. What is the probability that a randomly selected parcel arrived late?