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**Section 5.5 - Independent Events**

Andy (A), Bob (B), and Charlie (C) take turns rolling a fair die. Find the probability that Charlie rolls a five before Andy or Bob.

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**Section 5.5 - Independent Events**

Andy (A), Bob (B), and Charlie (C) take turns rolling a fair die. Find the probability that Charlie rolls a five before Andy or Bob.

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**Section 5.5 - Independent Events**

Andy (A), Bob (B), and Charlie (C) take turns rolling a fair die. Find the probability that Charlie rolls a five before Andy or Bob.

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**Section 5.5 - Independent Events**

P37. Suppose you select one person at random from the Titanic passengers and crew. Use the definition of independent events to determine whether the events “didn’t survive” and “male” are independent. Are any two events in this table independent? Display 5.39 Gender Male Female Total Survived? Yes 367 344 711 No 1364 126 1490 1731 470 2201

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**Section 5.5 - Independent Events**

P37. Determine whether the events “didn’t survive” and “male” are independent. Are any two events in this table independent? Display 5.39 Gender Male Female Total Survived? Yes 0.2120 0.7319 0.3230 No 0.7880 0.2681 0.6770 1.0000

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**Section 5.5 - Independent Events**

P38. Suppose you draw a card at random from a standard deck. Use the definition of independent events to determine which pairs of events are independent. Getting a heart; Getting a jack Getting a heart; Getting a red card Getting a 7; Getting a heart

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**Section 5.5 - Independent Events**

P38. Suppose you draw a card at random from a standard deck. Use the definition of independent events to determine which pairs of events are independent. Getting a heart; Getting a jack Getting a heart; Getting a red card Getting a 7; Getting a heart

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**Section 5.5 - Independent Events**

P39. About 42% of people have type O blood. Suppose you select two people at random and check whether they have type O blood. Make a table to show all possible results. What is the probability that exactly one of the two people has type O blood? Make a tree diagram that illustrates the situation.

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**Section 5.5 - Independent Events**

P39. Make a table to show all possible results. Second Person O Not O Total First Person 0.1764 0.2436 0.4200 0.3364 0.5800 1.0000

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**Section 5.5 - Independent Events**

P39. Make a table to show all possible results. What is the probability that exactly one of the two people has type O blood? Second Person O Not O Total First Person 0.1764 0.2436 0.4200 0.3364 0.5800 1.0000

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**Section 5.5 - Independent Events**

P39. Make a table to show all possible results. What is the probability that exactly one of the two people has type O blood? Second Person O Not O Total First Person 0.1764 0.2436 0.4200 0.3364 0.5800 1.0000

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**Section 5.5 - Independent Events**

P39. Make a tree diagram that illustrates the situation.

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**Section 5.5 - Independent Events**

P40. Suppose you select ten people at random. Using the information in P39, find the probability that at least one of them has type O blood at least one of them doesn’t have type O blood

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**Section 5.5 - Independent Events**

P40. Suppose you select ten people at random. Using the information in P39, find the probability that at least one of them has type O blood Second Person O Not O Total First Person 0.1764 0.2436 0.4200 0.3364 0.5800 1.0000

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**Section 5.5 - Independent Events**

P40. Suppose you select ten people at random. Using the information in P39, find the probability that at least one of them doesn’t have type O blood Second Person O Not O Total First Person 0.1764 0.2436 0.4200 0.3364 0.5800 1.0000

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**Section 5.5 - Independent Events**

P41. After taking college placement tests, freshmen sometimes are required to repeat high school work. Such work is called “remediation” and does not count toward a college degree. About 11% of college freshmen have to take a remedial course in reading. Suppose you select two freshmen at random and check to see if they have to take remedial reading. Find the probability that both freshmen have to take remedial reading. Show how to use a two-way table to find the probability that exactly one freshman needs to take remedial reading Show how to use a tree-diagram to answer the question in part b.

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**Section 5.5 - Independent Events**

P41a. About 11% of college freshmen have to take a remedial course in reading. Suppose you select two freshmen at random and check to see if they have to take remedial reading. Find the probability that both freshmen have to take remedial reading.

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**Section 5.5 - Independent Events**

P41b. Show how to use a two-way table to find the probability that exactly one freshman needs to take remedial reading Second Freshman Needs RR Doesn’t Total First Freshman 0.0121 0.0979 0.1100 0.7921 0.8900 1.0000

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**Section 5.5 - Independent Events**

P41c. Show how to use a tree-diagram to answer the question in part b.

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**Section 5.5 - Independent Events**

E58. Use the definition of independent events to determine which of these pairs of events are independent when you roll two dice. Rolling doubles; rolling a sum of 8 Rolling a sum of 8; getting a 2 on the first die rolled Rolling a sum of 7; getting a 1 on the first die rolled Rolling doubles; rolling a sum of 7 Rolling a 1 on the first die; rolling a 1 on the second die

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**Section 5.5 - Independent Events**

E58. Use the definition of independent events to determine which of these pairs of events are independent when you roll two dice. Rolling doubles; rolling a sum of 8 Rolling a sum of 8; getting a 2 on the first die rolled Rolling a sum of 7; getting a 1 on the first die rolled Rolling doubles; rolling a sum of 7 Rolling a 1 on the first die; rolling a 1 on the second die

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**Section 5.5 - Independent Events**

E64. Schizophrenia affects 1% of the U.S. population and tends to first appear between the ages of 18 and 25. Today, schizophrenia accounts for the fifth highest number of years of work lost due to disability among Americans aged Fewer than 30% of people with this illness are currently employed. Suppose you select a person from the U.S. population at random. State whether each question can be answered using the information given. If the question can be answered, answer it.

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**Section 5.5 - Independent Events**

E64. Schizophrenia affects 1% of the U.S. population and tends to first appear between the ages of 18 and 25. Today, schizophrenia accounts for the fifth highest number of years of work lost due to disability among Americans ages Fewer than 30% of people with this illness are currently employed. What is the probability that the person is schizophrenic? What is the probability that the person is unemployed? What is the probability that the person is unemployed and schizophrenic? What is the probability that the person is unemployed or schizophrenic? What is the probability that the person is unemployed given that he is schizophrenic? What is the probability that he is schizophrenic given that he is unemployed?

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**Section 5.5 - Independent Events**

E64. Schizophrenia affects 1% of the U.S. population and tends to first appear between the ages of 18 and 25. Today, schizophrenia accounts for the fifth highest number of years of work lost due to disability among Americans ages Fewer than 30% of people with this illness are currently employed. Schizophrenic? Yes No Total Employed? 0.003 0.007 0.010 0.990 1.000

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**Section 5.5 - Independent Events**

E64. Schizophrenia What is the probability that the person is schizophrenic? What is the probability that the person is unemployed? What is the probability that the person is unemployed and schizophrenic? What is the probability that the person is unemployed or schizophrenic? What is the probability that the person is unemployed given that he is schizophrenic? What is the probability that he is schizophrenic given that he is unemployed? Schizophrenic? Yes No Total Employed? 0.003 0.007 0.010 0.990 1.000

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**Section 5.5 - Independent Events**

E64. Schizophrenia affects 1% of the U.S. population and tends to first appear between the ages of 18 and 25. Today, schizophrenia accounts for the fifth highest number of years of work lost due to disability among Americans ages Fewer than 30% of people with this illness are currently employed. What is the probability that the person is schizophrenic? 0.01 What is the probability that the person is unemployed? Can’t be determined What is the probability that the person is unemployed and schizophrenic? 0.007 What is the probability that the person is unemployed or schizophrenic? Can’t be determined What is the probability that the person is unemployed given that he is schizophrenic? More than 0.70 What is the probability that he is schizophrenic given that he is unemployed? Can’t be determined

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**Section 5.5 - Independent Events**

E66. A committee of three students is to be formed from six juniors and five seniors. If three names are drawn at random, what is the probability that the committee will consist of all juniors or all seniors?

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**Section 5.5 - Independent Events**

E66. A committee of three students is to be formed from six juniors and five seniors. If three names are drawn at random, what is the probability that the committee will consist of all juniors or all seniors?

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**Section 5.5 - Independent Events**

E66. A committee of three students is to be formed from six juniors and five seniors. If three names are drawn at random, what is the probability that the committee will consist of all juniors or all seniors?

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**Section 5.5 - Independent Events**

E70. Joaquin computes the probability that if two dice are rolled, then exactly one die will show a 4 as: Joaquin is wrong. What did he forget?

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**Section 5.5 - Independent Events**

E70. Joaquin computes the probability that if two dice are rolled, then exactly one die will show a 4 as: Joaquin is wrong. What did he forget?

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**Section 5.5 - Independent Events**

sometimes is called the law of total probability. Prove the law of total probability Use this law to find P(Dodgers win) from the information in Display 5.51. Prove Bayes’ Theorem: Use Bayes’ Theorem to find P(Dodgers win|day game)

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**Section 5.5 - Independent Events**

sometimes is called the law of total probability. Prove the law of total probability B B’ A A B A B’ A’

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**Section 5.5 - Independent Events**

sometimes is called the law of total probability. Use this law to find P(Dodgers win) from the information in Display 5.51. Won the Game? Yes No Total Time of Game Day 11 10 21 Night 30 27 57 41 37 78

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**Section 5.5 - Independent Events**

sometimes is called the law of total probability. Use this law to find P(Dodgers win) from the information in Display 5.51. Won the Game? Yes No Total Time of Game Day 11 10 21 Night 30 27 57 41 37 78

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**Section 5.5 - Independent Events**

E75. Prove Bayes’ Theorem

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**Section 5.5 - Independent Events**

E75. Use Bayes’ Theorem to find P(Dodgers win|day game) Won the Game? Yes No Total Time of Game Day 11 10 21 Night 30 27 57 41 37 78

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**Section 5.5 - Independent Events**

E75. Use Bayes’ Theorem to find P(Dodgers win|day game) Won the Game? Yes No Total Time of Game Day 11 10 21 Night 30 27 57 41 37 78

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**Section 5.5 - Independent Events**

E75. Use Bayes’ Theorem to find P(Dodgers win|day game) Won the Game? Yes No Total Time of Game Day 11 10 21 Night 30 27 57 41 37 78

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