Exercise Set 6.5: Conditional Probability and Independent Events Find the following, given that P ( A) = 0.71 , Answer the following. P ( B ) = 0.53 , and P ( A B ) = 0.28 . 25. An experiment consists of rolling 2 dice and observing the number that lands on the uppermost surface. Find the probability that the sum of the numbers is 10, given that one of the numbers is a 4. ( ) P(A B ) P (B A ) 1. P ( B A ) 4. P B A c 2. P ( A B ) 5. ( 3. P A B c ) 6. c c c c Find the following, given that P ( A) = 0.56 , P ( B ) = 0.63 , and P ( A B ) = 0.41 . 7. P ( A B ) 8. P ( B A ) ( 9. P B A c ( ) 11. P ( B A ) 12. P ( A B ) 10. P A B c ) c c c c Find the following, given that c P ( A B c ) = 0.82 , P ( A B ) = 0.24 and ( ) P A B c = 0.33 . 13. P ( B A ) 14. P ( A B ) ( 15. P A B c ( ) 17. P ( A B ) 18. P ( B A ) 16. P B A c ) c c c c Find the following, given that c P A B c = 0.75 , P ( A B ) = 0.04 , and ( ) P ( A B ) = 0.65 . c 19. P ( A B ) 20. P ( B A ) ( 21. P B A c ( ) 23. P ( B A ) 24. P ( A B ) 22. P A B c ) c c c c 26. An experiment consists of rolling 2 dice and observing the number that lands on the uppermost surface. Find the probability that the sum of the numbers is 6, given that one of the numbers is a 2. 27. An experiment consists of rolling 2 dice and observing the number that lands on the uppermost surface. Find the probability that the one of the numbers is a 4, given that the sum of the numbers is odd. 28. An experiment consists of rolling 2 dice and observing the number that lands on the uppermost surface. Find the probability that the one of the numbers is a 5, given that the sum of the numbers is even. 29. Suppose that you draw two cards from a well-shuffled deck of 52 playing cards without replacement. What is the probability that the second card is an ace, given that the first card is not an ace? 30. Suppose that you draw two cards from a well-shuffled deck of 52 playing cards without replacement. What is the probability that the second card is an ace, given that the first card is an ace? 31. Suppose that you draw two cards from a well-shuffled deck of 52 playing cards without replacement. What is the probability that the second card is a spade, given that the first card is not a spade? 32. Suppose that you draw two cards from a well-shuffled deck of 52 playing cards without replacement. What is the MATH 1313 Finite Mathematics with Applications 403 Exercise Set 6.5: Conditional Probability and Independent Events probability that the second card is red, given that the first card is red? 33. An experiment consists of drawing two marbles from an urn and observing the color. There are 8 red marbles and 7 white marbles in the urn. Find the probability that both marbles are red, assuming: A. The first marble is returned to the urn before the second marble is drawn. B. The first marble is not returned to the urn before the second marble is drawn. 34. An experiment consists of drawing two marbles from an urn and observing the color. There are 14 red marbles and 9 white marbles in the urn. Find the probability that both marbles are red, assuming: A. The first marble is returned to the urn before the second marble is drawn. B. The first marble is not returned to the urn before the second marble is drawn. 35. An experiment consists of selecting an urn and then removing a marble from the urn. There are 12 red marbles and 8 blue marbles in Urn 1, and there are 9 red marbles and 15 blue marbles in Urn 2. Assume the urns are equally likely to be selected. Find the probability that: A. The marble is blue. B. The marble is red. C. Urn 2 is chosen and a blue marble is drawn. D. A red marble is drawn from Urn 2. E. A red marble is drawn, given that Urn 2 is selected. F. A blue marble is drawn, given that Urn 1 is selected. 36. An experiment consists of selecting an urn and then removing a marble from the urn. There are 8 red marbles and 22 blue marbles in Urn 1, and there are 14 red 404 marbles and 5 blue marbles in Urn 2. The probability that Urn 1 is selected is 0.4 and the probability that Urn 2 is selected is 0.6. Find the probability that: A. B. C. D. The marble is blue. The marble is red. A blue marble is drawn from Urn 2. Urn 2 is chosen and a red marble is drawn. E. A red marble is drawn, given that Urn 2 is selected. F. A blue marble is drawn, given that Urn 1 is selected. 37. An experiment consists of selecting two marbles from an urn, in succession and without replacement. The urn contains 14 green marbles and 11 yellow marbles. Find the probability that: A. B. C. D. E. The second marble is green. The second marble is yellow. Both marbles are yellow. Both marbles are green. The second marble is green, given that the first marble is green. F. The second marble is yellow, given that the first marble is green. 38. An experiment consists of selecting two marbles from an urn, in succession and without replacement. The urn contains 17 green marbles and 16 yellow marbles. Find the probability that: A. B. C. D. E. The second marble is green. The second marble is yellow. Both marbles are yellow. Both marbles are green. The second marble is green, given that the first marble is green. F. The second marble is yellow, given that the first marble is green. University of Houston Department of Mathematics Exercise Set 6.5: Conditional Probability and Independent Events 39. A survey of 1000 people was conducted and the respondents were asked if they were colorblind. The results are summarized in the following table: A single card is drawn randomly from a standard deck. Use the definition of independent events to determine if events A and B are independent. Male Female 75 23 Colorblind 491 Not colorblind 411 Recall: The definition of independent events states that events A and B are independent if and only if P ( A ) = P ( A | B ) . Find the probability that: A. The person surveyed is colorblind. B. The person surveyed is male and colorblind. C. The person surveyed is female and not colorblind. D. The person is female, given that the person is colorblind. E. The person is male, given that the person is not colorblind. 40. A survey of 1000 people was conducted and the respondents were asked whether or not they smoke. The results are summarized in the following table: Male Female 127 66 Smoker 404 Non-smoker 403 Find the probability that: A. The person surveyed is a smoker. B. The person surveyed is male and a nonsmoker. C. The person surveyed is female and a smoker. D. The person is female, given that the person is a non-smoker. E. The person is male, given that the person is a smoker. 41. Event A: Drawing a heart Event B: Drawing an Ace 42. Event A: Drawing a King Event B: Drawing a face card 43. Event A: Drawing a black card Event B: Drawing a heart 44. Event A: Drawing a Queen Event B: Drawing a red card 45. Event A: Drawing a diamond Event B: Drawing a red card 46. Event A: Drawing a 7 Event B: Drawing a Jack A single card is drawn randomly from a standard deck. Use the product rule test to determine if events A and B are independent. Recall: The product rule test states that events A and B are independent if and only if P ( A B ) = P ( A) P ( B ) . 47. Event A: Drawing a 10 Event B: Drawing an black card 48. Event A: Drawing a heart Event B: Drawing a red card 49. Event A: Drawing a face card Event B: Drawing a Jack 50. Event A: Drawing a spade Event B: Drawing a 3 MATH 1313 Finite Mathematics with Applications 405 Exercise Set 6.5: Conditional Probability and Independent Events Use the given information to determine if events A and B are independent. 51. P ( A) = 0.9, P ( B ) = 0.3, P ( A B ) = 0.27 52. P ( A) = 0.4, P ( B ) = 0.6, P ( A B ) = 0.20 53. P ( A) = 0.5, P ( B ) = 0.9, P ( A B ) = 0.56 54. P ( A) = 0.7, P ( B ) = 0.6, P ( A B ) = 0.42 In problems 55-58, assume that events A and B are independent and find the following: A. P ( A B) B. P ( A B) 55. P ( A ) = 0.85, P ( B ) = 0.71 56. P ( A) = 0.39, P ( B ) = 0.63 57. P ( A) = 0.67, P ( B ) = 0.55 58. P ( A) = 0.49, P ( B ) = 0.61 Answer the following. 59. An experiment consists of tossing a fair coin twice and observing whether the coin lands with heads or tails facing up. Define E as the event that the coin lands with heads facing up on the first toss. Define F as the event that the coin lands with heads facing up on the second toss. Determine if the two events are independent. 60. An experiment consists of selecting a card from a well-shuffled deck of 52 playing cards, replacing it, and then selecting a card from the deck. Define E as the event of selecting a jack on the first draw, and define F as the event of selecting a red card on the second draw. Determine if the events are independent. 61. A company owns three photocopiers. The probabilities that each of the copiers will break down on a given day are 0.04, 0.02 and 0.018 respectively. Assuming independence, find the probability that all three copiers: 406 A. Break down on a given day. B. Are working on a given day. 62. A company owns three photocopiers. The probabilities that each of the copiers will break down on a given day are 0.071, 0.017 and 0.025 respectively. Assuming independence, find the probability that all three copiers: A. Break down on a given day. B. Are working on a given day. 63. Adam, Becca and Carl all have the same medical treatment. The probability that Adam will experience a full recovery is 0.82, the probability that Becca will experience a full recovery is 0.91, and the probability that Carl will experience a full recovery is 0.88. Assuming independence, find the probability that: A. Both Adam and Carl fully recover. B. Both Adam and Becca fully recover. C. Both Becca and Carl fully recover. D. All three fully recover. E. None of the three fully recover. 64. David, Edie and Francesca all have the same medical treatment. The probability that David will experience a full recovery is 0.59; the probability that Edie will experience a full recovery is 0.81; the probability that Francesca will experience a full recovery is 0.77. Assuming independence, find the probability that: A. Both David and Edie fully recover. B. Both Edie and Francesca fully recover. C. Both David and Francesca fully recover. D. All three fully recover. E. None of the three fully recover. University of Houston Department of Mathematics Math 1313 Homework 21 Section 6.5 1. The choices for problem number 10 from the book are given below. a. 0.4054 b. 0.1500 c. 0.5946 d. 0.7321 e. 0.2381 2. The choices for problem number 24 from the book are given below. a. 0.6500 b. 0.1852 c. 0.0580 d. 0.4421 e. 0.3742 3. The choices for problem number 28 from the book are given below. a. 0.7333 b. 0.2778 c. 0.7222 d. 0.5421 e. 0.3241 4. The choices for problem number 38 part d from the book are given below. a. 0.5006 b. 0.0682 c. 0.6123 d. 0.2872 e. 0.2576 5. The choices for problem number 40 part e from the book are given below. a. 0.5806 b. 0.1270 c. 0.8183 d. 0.6580 e. 0.4138 Use the following problems to answer questions 6 - 8 Of 2,000 students surveyed, 85% indicated that they commute to school each day. Of those that commute to school each day, 88% spend more than $25 a week on gasoline. Of those that do not commute to school each day, 95% indicated that they do not spend more than $25 a week on gasoline. A student surveyed is chosen at random. 6. What is the probability that this student does not commute to school each day and does not spend more than $25 a week on gas? a. 0.8075 b. 0.1425 Math 1313 Homework 21 Section 6.5 c. 0.0500 d. 0.1200 e. 0.1900 7. What is the probability that this student does spend more than $25 a week on gas? a. 0.2445 b. 0.7555 c. 0.6740 d. 0.3260 e. 0.4532 8. What is the probability that this student does spend more than $25 a week on gas, given that they commute to school each day? a. 0.1200 b. 0.8300 c. 0.0500 d. 0.9500 e. 0.8800 Use the following problems to answer question 9 and 10. There are 300 seniors in Jefferson High School, of which 140 are males. It is known that 70% of the males and 65% of the females have their driver's license. If a student is selected at random from this senior class, what is the probability that the student is: 9. A female and has a driver's license? a. 0.6500 b. 0.3733 c. 0.3467 d. 0.2478 e. 0.5863 10. That they do not have a driver's license? a. 0.5234 b. 0.3267 c. 0.4286 d. 0.3497 e. 0.6108