Exercise Set 6.6: Bayes' Theorem Use the tree diagram to find each probability in problems 1-6. Event D is defined to be the event that an item is defective. Answer the following. 13. An experiment consists of drawing two marbles from an urn in succession and without replacement and observing the color of the marbles. There are 14 black marbles and 16 green marbles in the urn. Find the probability that: A. The second marble is green. B. The second marble is black. C. The first marble is black, given that the second marble is black. D. The first marble is black, given that the second marble is green. E. The second marble is green, given that the first marble was black. 1. P ( D ) 2. P ( D c ) 4. P ( B D ) 5. P C D c ( 14. An experiment consists of drawing two marbles from an urn in succession and without replacement and observing the color of the marbles. There are 24 yellow marbles and 16 blue marbles in the urn. Find the probability that: 3. P ( A D ) ) ( 6. P A D c ) Use the tree diagram to find each probability in problems 7-12. Event D is defined to be the event that an item is defective. A. The second marble is blue. B. The second marble is yellow. C. The first marble is yellow, given that the second marble is yellow. D. The first marble is yellow, given that the second marble is blue. E. The second marble is yellow, given that the first marble was blue. 15. An experiment consists of selecting one of two urns and then removing a marble from the urn. Urn I contains 11 red marbles and 13 green marbles. Urn II contains 15 red marbles and 9 green marbles. Assume that the urns are equally likely to be selected. Find the probability that: 7. P ( D c ) 8. P ( D ) 10. P ( A D ) 11. P A D c 418 ( 9. P ( B D ) ) ( 12. P C D c ) A. Urn II and a green marble are selected. B. A red marble is selected. C. A green marble is selected. Continued on the next page... University of Houston Department of Mathematics Exercise Set 6.6: Bayes' Theorem D. A marble was selected from Urn I, if it is known that the marble is green. E. The marble was selected from Urn II, if it is known that the marble is red. F. The marble is red, if it is known that the marble was selected from Urn II. 16. An experiment consists of selecting one of two urns and then removing a marble from the urn. Urn I contains 19 violet marbles and 13 white marbles. Urn II contains 17 violet marbles and 8 white marbles. Assume that the urns are equally likely to be selected. Find the probability that: A. B. C. D. Urn II and a white marble are selected. A violet marble is selected. A white marble is selected. A marble was selected from Urn I, if it is known that the marble is violet. E. The marble was selected from Urn II, if it is known that the marble is white. F. The marble is white, if it is known that the marble was selected from Urn II. 17. An experiment consists of selecting one of three urns and then selecting a marble from the urn. Urn I contains 8 blue marbles and 7 white marbles. Urn II contains 10 blue marbles and 6 white marbles. Urn III contains 7 blue marbles and 11 white marbles. The probability that Urn I will be selected is 0.40, the probability that Urn II will be selected is 0.30, and the probability that Urn III will be selected is 0.30. Find the probability that: A. B. C. D. Urn I and a blue marble are selected. A white marble is selected. A blue marble is selected. The marble was selected from Urn II, if it is known that the marble is blue. E. The marble was selected from Urn III, if it is known that the marble is white. F. The marble is white, if it is known that the marble was selected from Urn I. MATH 1313 Finite Mathematics with Applications 18. An experiment consists of selecting one of three urns and then selecting a marble from the urn. Urn I contains 9 yellow marbles and 7 brown marbles. Urn II contains 10 yellow marbles and 6 brown marbles. Urn III contains 7 yellow marbles and 11 brown marbles. The probability that Urn I will be selected is 0.42, the probability that Urn II will be selected is 0.45, and the probability that Urn III will be selected is 0.13. Find the probability that: A. B. C. D. Urn I and a yellow marble are selected. A brown marble is selected. A yellow marble is selected. The marble was selected from Urn II, if it is known that the marble is yellow. E. The marble was selected from Urn III, if it is known that the marble is brown. F. The marble is brown, if it is known that the marble was selected from Urn I. 19. A game consists of selecting one of three doors, opening it, and then tossing the coin that is behind the door. Door I is more likely to be selected than the other two, with a probability of 0.58. The other two doors each have a probability of 0.21 of being selected. Behind Door I is a fair coin. Behind Door II is a biased coin with probability of 0.66 that the coin lands on heads. Behind Door III is a biased coin, with probability of 0.28 that the coin lands on tails. A. Suppose that you win a prize if the coin lands on heads. What is the probability that you win a prize? B. If the coin lands on tails, what is the probability that it was behind Door III? C. If the coin lands on heads, what is the probability that it was behind Door II? D. What is the probability that Door I is selected and the coin lands on heads? E. What is the probability that the coin lands on tails, if it is known that Door II was selected? 419 Exercise Set 6.6: Bayes' Theorem 20. A game consists of selecting one of three doors, opening it, and then tossing the coin that is behind the door. Door I is more likely to be selected than the other two, with a probability of 0.64. The other two doors each have a probability of 0.18 of being selected. Behind Door I is a fair coin. Behind Door II is a biased coin with probability of 0.75 that the coin lands on heads. Behind Door III is a biased coin, with probability of 0.33 that the coin lands on tails. A. Suppose that you win a prize if the coin lands on heads. What is the probability that you win a prize? B. If the coin lands on tails, what is the probability that it was behind Door III? C. If the coin lands on heads, what is the probability that it was behind Door II? D. What is the probability that Door I is selected and the coin lands on heads? E. What is the probability that the coin lands on tails, if it is known that Door II was selected? 21. A group of high school students were surveyed about their post-high school education plans. Of the 500 students surveyed, 371 plan to go to a four-year college and 129 plan to go to a community college. Among the students who plan to attend a four-year college, 133 plan to major in business. Among the students who plan to attend a two-year college, 35 plan to major in business. Find the probability that a randomly selected surveyed student: A. Plans to major in business. B. Plans to major in something other than business. C. Plans to attend a four-year college and major in something other than business. D. Plans to attend a community college and major in business. Continued in the next column... 420 E. Plans to attend a four-year college, given that the student plans to major in business. F. Plans to attend a community college, given that the student plans to major in something other than business. G. Plans to major in business, given that the student plans to attend a community college. 22. A group of high school students were surveyed about their post-high school education plans. Of the 500 students who were surveyed, 309 plan to go to a fouryear college and 191 plan to go to a community college. Among the students who plan to attend a four-year college, 103 plan to major in education. Among the students who plan to attend a two-year college, 21 plan to major in education. Find the probability that a randomly selected surveyed student: A. Plans to major in education. B. Plans to major in something other than education. C. Plans to attend a four-year college and major in something other than education. D. Plans to attend a community college and major in education. E. Plans to attend a four-year college, given that the student plans to major in education. F. Plans to attend a community college, given that the student plans to major in something other than education. G. Plans to major in education, given that the student plans to attend a community college. 23. Celiac disease is present in about 0.75% of the American population. A combination of blood tests accurately diagnoses Celiac disease in people who have the disease 90% of the time. The tests give false positive University of Houston Department of Mathematics Exercise Set 6.6: Bayes' Theorem results for 6% of people who do not have Celiac disease. Find the probability that: A. The blood tests show that a person does not have Celiac disease. B. The blood tests show that a person has Celiac disease. C. A person has Celiac disease and the test results were positive. D. A person has Celiac disease, given that the test results were positive. E. A person does not have Celiac disease, given that the test results were positive. F. The test results were negative, given that a person does not have Celiac disease. 24. About 8.3% of the American population has diabetes. A combination of blood tests accurately diagnoses diabetes 99.3% of the time. The tests give false positive results for 1.2% of people who do not have the diabetes. Find the probability that: A. The blood tests show that a person does not have diabetes. B. The blood tests show that a person has diabetes. C. A person has diabetes and the test results were positive. D. A person has diabetes, given that the test results were positive. E. A person does not have diabetes, given that the test results were positive. F. The test results were negative, given that a person does not have diabetes. 25. A company buys one of its components from three different suppliers: 20% of the components are purchased from Supplier A, 45% of the components are purchased from Supplier B, and 35% of the components are purchased from Supplier C. The company knows that some of the components will be defective: 4.5% of the components from Supplier A are defective, 1.8% of the MATH 1313 Finite Mathematics with Applications components from Supplier B are defective and 2.3% of the components from Supplier C are defective. Find the probability that a randomly selected component: A. Is defective. B. Is defective and came from Supplier B. C. Is not defective and came from Supplier C. D. Is defective, if it is known that the component came from Supplier A. E. Came from Supplier A, if it is known that the component is defective. F. Came from Supplier B, if it is known that the component is defective. 26. A company buys one of its components from three different suppliers: 38% of the components are purchased from Supplier A, 20% of the components are purchased from Supplier B, and 42% of the components are purchased from Supplier C. The company knows that some of the components will be defective: 2.4% of the components from Supplier A are defective, 5.1% of the components from Supplier B are defective and 1.3% of the components from Supplier C are defective. Find the probability that a randomly selected component: A. Is defective. B. Is defective and came from Supplier B. C. Is not defective and came from Supplier C. D. Is defective, if it is known that the component came from Supplier A. E. Came from Supplier A, if it is known that the component is defective. F. Came from Supplier B, if it is known that the component is defective. 421 Math 1313 Homework 22 Section 6.6 1. The choices for problem number 10 from the book are given below. a. 0.3108 b. 0.3902 c. 0.7812 d. 0.4236 e. 0.0112 2. The choices for problem number 12 from the book are given below. a. 0.2908 b. 0.0304 c. 0.4137 d. 0.0840 e. 0.1403 3. The choices for problem number 16 part d from the book are given below. a. 0.5332 b. 0.4406 c. 0.5278 d. 0.3863 e. 0.4661 4. The choices for problem number 24 part b from the book are given below. a. 0.8802 b. 0.9066 c. 0.3415 d. 0.0934 e. 0.6685 5. The choices for problem number 24 part e from the book are given below. a. 0.1178 b. 0.4571 c. 0.1623 d. 0.3201 e. 0.0427 Use the following problem to answer questions 6 and 7. Urn 1 contains 14 green and 15 yellow marbles. Urn 2 contains 11 green and 8 yellow marbles. An experiment consists of choosing one of two urns at random then drawing a marble from the chosen urn. Urn 1 is more likely to be chosen than Urn 2 with probability 0.57. 6. What is the probability that a green marble was chosen? a. 0.5155 b. 0.3768 c. 0.4859 d. 0.5241 e. 0.4828 Math 1313 Homework 22 Section 6.6 7. What is the probability that Urn 1 was chosen, if it is known that a yellow marble was drawn? a. 0.6195 b. 0.7223 c. 0.5874 d. 0.4698 e. 0.0841 Use the following problem to answer questions 8 and 9. A television manufacturer has three locations where it is finally assembled. Plants A, B, and C supply 65%, 15%, and 20% respectively of the televisions used by the manufacture. Quality control has determined that 1.5% produced by Plant A are defective, while plants B and C have 2.5% defective products. 8. What is the probability of finding a non-defective television? a. 0.0150 b. 0.9054 c. 0.9815 d. 0.0204 e. 0.9746 9. What is the probability that the television was manufactured by Plant B, given it was defective? a. 0.3846 b. 0.2027 c. 0.2985 d. 0.4975 e. 0.1493 10. At a carnival you win a prize if you get a heads, you must first choose a coin. There is a fair and a biased coin, while choosing each coin is equally likely, the biased coin has a 78% of landing tails. What is the probability of choosing the biased coin if you won a prize. a. 0.2146 b. 0.4084 c. 0.3421 d. 0.3056 e. 0.2670