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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.

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Presentation on theme: "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."— Presentation transcript:

1 Section 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.

2 Section 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.

3 Section 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.

4 Section 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 MaleFemaleTotal Survived? Yes No Total

5 Section 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 MaleFemaleTotal Survived? Yes No Total1.0000

6 Section 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. a.Getting a heart; Getting a jack b.Getting a heart; Getting a red card c.Getting a 7; Getting a heart

7 Section 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. a.Getting a heart; Getting a jack b.Getting a heart; Getting a red card c.Getting a 7; Getting a heart

8 Section 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. a.Make a table to show all possible results. b.What is the probability that exactly one of the two people has type O blood? c.Make a tree diagram that illustrates the situation.

9 Section Independent Events P39. Make a table to show all possible results. Second Person ONot OTotal First Person O Not O Total

10 Section 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 ONot OTotal First Person O Not O Total

11 Section 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 ONot OTotal First Person O Not O Total

12 Section Independent Events P39. Make a tree diagram that illustrates the situation.

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

14 Section Independent Events P40. Suppose you select ten people at random. Using the information in P39, find the probability that a.at least one of them has type O blood Second Person ONot OTotal First Person O Not O Total

15 Section Independent Events P40. Suppose you select ten people at random. Using the information in P39, find the probability that b.at least one of them doesn’t have type O blood Second Person ONot OTotal First Person O Not O Total

16 Section 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. a.Find the probability that both freshmen have to take remedial reading. b.Show how to use a two-way table to find the probability that exactly one freshman needs to take remedial reading c.Show how to use a tree-diagram to answer the question in part b.

17 Section 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.

18 Section 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 RRDoesn’tTotal First Freshman Needs RR Doesn’t Total

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

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

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

22 Section 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.

23 Section 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. a.What is the probability that the person is schizophrenic? b.What is the probability that the person is unemployed? c.What is the probability that the person is unemployed and schizophrenic? d.What is the probability that the person is unemployed or schizophrenic? e.What is the probability that the person is unemployed given that he is schizophrenic? f.What is the probability that he is schizophrenic given that he is unemployed?

24 Section 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? YesNoTotal Employed? Yes0.003 No0.007 Total

25 Section Independent Events E64. Schizophrenia a.What is the probability that the person is schizophrenic? b.What is the probability that the person is unemployed? c.What is the probability that the person is unemployed and schizophrenic? d.What is the probability that the person is unemployed or schizophrenic? e.What is the probability that the person is unemployed given that he is schizophrenic? f.What is the probability that he is schizophrenic given that he is unemployed? Schizophrenic? YesNoTotal Employed? Yes0.003 No0.007 Total

26 Section 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. a.What is the probability that the person is schizophrenic? 0.01 b.What is the probability that the person is unemployed? Can’t be determined c.What is the probability that the person is unemployed and schizophrenic? d.What is the probability that the person is unemployed or schizophrenic? Can’t be determined e.What is the probability that the person is unemployed given that he is schizophrenic? More than 0.70 f.What is the probability that he is schizophrenic given that he is unemployed? Can’t be determined

27 Section 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?

28 Section 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?

29 Section 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?

30 Section 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?

31 Section 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?

32 Section Independent Events E75. sometimes is called the law of total probability. a.Prove the law of total probability b.Use this law to find P(Dodgers win) from the information in Display c.Prove Bayes’ Theorem: d.Use Bayes’ Theorem to find P(Dodgers win|day game)

33 Section Independent Events E75. sometimes is called the law of total probability. Prove the law of total probability BB’ A A  BA  B’ A’

34 Section Independent Events E75. sometimes is called the law of total probability. Use this law to find P(Dodgers win) from the information in Display Won the Game? YesNoTotal Time of Game Day Night Total413778

35 Section Independent Events E75. sometimes is called the law of total probability. Use this law to find P(Dodgers win) from the information in Display Won the Game? YesNoTotal Time of Game Day Night Total413778

36 Section Independent Events E75. Prove Bayes’ Theorem

37 Section Independent Events E75. Use Bayes’ Theorem to find P(Dodgers win|day game) Won the Game? YesNoTotal Time of Game Day Night Total413778

38 Section Independent Events E75. Use Bayes’ Theorem to find P(Dodgers win|day game) Won the Game? YesNoTotal Time of Game Day Night Total413778

39 Section Independent Events E75. Use Bayes’ Theorem to find P(Dodgers win|day game) Won the Game? YesNoTotal Time of Game Day Night Total413778


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