.PDF

# Homework 3 _SOLUTION

1. For the data given in Homework 1, Problem 3, answer the following questions:(a) Given that a student gets problem 1 incorrect, what is the probability he gets Problem 2 incorrect? (b) Given that a student gets problem 1 correct, what is the probability he gets Problem 2 incorrect? (c) Given that a student gets both problem 1 and 2 incorrect, what is the probability he gets Problem 3 incorrect? (d) Using the deﬁnition for the independence of two events, are the events “student gets problem 1 incorrect” and “student gets problem 2 incorrect” independent? (Show your work, don’t just answer yes or no.)

2. Walpole Exercise 2.92. Copied here: Suppose the diagram of an electrical system is given in Figure 2.10. What is the probability the system works? Assume the components fail independently. Neal’s note: Figure 2.10 shows a system which, in order to work, must have the following statement true: Component A works AND Component D works AND (Component B works OR Component C works). Component A works with probability 0.95; Component B works with probability 0.7; Component C works with probability 0.8; and Component D works with probability 0.9.

3. A rare disease aﬀects 1 in 10,000 individuals in the population. A blood test for the disease is positive for 95% of people with the disease. However, 0.4% of people without the disease will also test positive.

(a) What is the likelihood that a person, selected at random from the population, has the rare disease and tests positive for it? (b) What is the likelihood that a person, selected at random from the population, does NOT have the rare disease and tests positive for it? (c) Given that a person (selected at random from the general population) tests positive, what is the probability that she has the disease? (d) Now assume that the doctor knows that 1 in 100 individuals with symptom G will have the rare disease, and he only orders the test when individuals have symptom G. Given that one of the doctor’s patients tests positive, what is the probability that she has the disease?

4. There is a 40% chance that the Seahawks win the Super Bowl. If they win, I have a 75% chance of writing an easy exam. If they lose, I have a 20% chance of writing an easy exam. You don’t know the outcome of the game (perhaps because you were so focused on studying), but you take the exam and it was NOT easy. What is the probability that the Seahawks won?

5. A family has three children. Assume that the sex of one child is independent from the others, and boy and girl are equally likely. Given that the family has one or more girls, what is the probability it has three girls?

ECE 3530 / CS 3130 Spring 2015 2

6. Walpole 2.99. Copied here: Supposethat the four inspectors at a ﬁlm factory are supposed to stamp the expiration date on each package of ﬁlm at the end of the assembly line. John, who stamps 20% of the packages, fails to stamp the expiration date once in every 200 packages; Tom, who stamps 60% of the packages, fails to stamp the expiration date once in every 100 packages; Jeﬀ, who stamps 15% of the packages, fails to stamp the expiration date once in every 90 packages; and Pat, who stamps 5% of the packages, fails to stamp the expiration date once in every 200 packages. If a customer complains that her package of ﬁlm does not show the expiration date, what is the probability that it was inspected by John?

7. Walpole 2.127. Copied here: There is a 50-50 chance that the queen carries the gene of hemophilia. If she is a carrier, then each prince has a 50-50 chance of having hemophilia, independently of the other princes. If the queen is not a carrier, the prince will not have the disease. Suppose the queen has had three princes, each without the disease. What is the probability the queen is a carrier?

2. Walpole Exercise 2.92. Copied here: Suppose the diagram of an electrical system is given in Figure 2.10. What is the probability the system works? Assume the components fail independently. Neal’s note: Figure 2.10 shows a system which, in order to work, must have the following statement true: Component A works AND Component D works AND (Component B works OR Component C works). Component A works with probability 0.95; Component B works with probability 0.7; Component C works with probability 0.8; and Component D works with probability 0.9.

3. A rare disease aﬀects 1 in 10,000 individuals in the population. A blood test for the disease is positive for 95% of people with the disease. However, 0.4% of people without the disease will also test positive.

(a) What is the likelihood that a person, selected at random from the population, has the rare disease and tests positive for it? (b) What is the likelihood that a person, selected at random from the population, does NOT have the rare disease and tests positive for it? (c) Given that a person (selected at random from the general population) tests positive, what is the probability that she has the disease? (d) Now assume that the doctor knows that 1 in 100 individuals with symptom G will have the rare disease, and he only orders the test when individuals have symptom G. Given that one of the doctor’s patients tests positive, what is the probability that she has the disease?

4. There is a 40% chance that the Seahawks win the Super Bowl. If they win, I have a 75% chance of writing an easy exam. If they lose, I have a 20% chance of writing an easy exam. You don’t know the outcome of the game (perhaps because you were so focused on studying), but you take the exam and it was NOT easy. What is the probability that the Seahawks won?

5. A family has three children. Assume that the sex of one child is independent from the others, and boy and girl are equally likely. Given that the family has one or more girls, what is the probability it has three girls?

ECE 3530 / CS 3130 Spring 2015 2

6. Walpole 2.99. Copied here: Supposethat the four inspectors at a ﬁlm factory are supposed to stamp the expiration date on each package of ﬁlm at the end of the assembly line. John, who stamps 20% of the packages, fails to stamp the expiration date once in every 200 packages; Tom, who stamps 60% of the packages, fails to stamp the expiration date once in every 100 packages; Jeﬀ, who stamps 15% of the packages, fails to stamp the expiration date once in every 90 packages; and Pat, who stamps 5% of the packages, fails to stamp the expiration date once in every 200 packages. If a customer complains that her package of ﬁlm does not show the expiration date, what is the probability that it was inspected by John?

7. Walpole 2.127. Copied here: There is a 50-50 chance that the queen carries the gene of hemophilia. If she is a carrier, then each prince has a 50-50 chance of having hemophilia, independently of the other princes. If the queen is not a carrier, the prince will not have the disease. Suppose the queen has had three princes, each without the disease. What is the probability the queen is a carrier?

You'll get a 97.6KB .PDF file.