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8.4 Expecting the Unexpected

8.4 Expecting the Unexpected
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  • 23. Bayes' Theorem

    For any two events with probabilities greater than 0,

    You have the following information about students at a college.

    • 49% of the students are male.

    • 11% of the students are nursing majors.

    • 9% of the nursing majors are male.

    What is the probability that a student is a nursing major given that the student is male?

    • Worked-Out Solution

      Event 1 is "the student is a nursing major."

      Event 2 is "the student is male."

      Event 2, given event 1 is "the nursing major is male."

      A set diagram might help you see what is going on. In the diagram, you can see that 9% of the 11% of the student population are male nursing majors. If you are given that the selected student is male, then you have restricted your total population to the 49% of the students who are male. So, given that a student is male, the probability that he is a nursing major is

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      _____     ______    ____      ______    ______  
     /  ___||  /_   _//  |  _ \\   /_   _//  /_   _// 
    | // __     -| ||-   | |_| ||   -| ||-     | ||   
    | \\_\ ||   _| ||_   | .  //    _| ||_    _| ||   
     \____//   /_____//  |_|\_\\   /_____//  /__//    
      `---`    `-----`   `-` --`   `-----`   `--`     
                                                      
    
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  • 24. Bayes' Theorem

    For any two events with probabilities greater than 0,

    You have the following information about students at a college.

    • 51% of the students are female
    • 10% of the students are history majors
    • 60% of the history majors are female

    What is the probability that a student is a history major given that the student is female?

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     ______      ___      _____     ______   _    _   
    |      \\   / _ \\   / ____||  /_   _// | || | || 
    |  --  //  / //\ \\ / //---`'   -| ||-  | || | || 
    |  --  \\ |  ___  ||\ \\___     _| ||_  | \\_/ || 
    |______// |_||  |_|| \_____||  /_____//  \____//  
    `------`  `-`   `-`   `----`   `-----`    `---`   
                                                      
    
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  • 25. Bayes' Theorem

    For any two events with probabilities greater than 0,

    You have the following information about voters in a local mayoral election.

    • 61% of voters were registered Republican.
    • 53% of voters voted Republican.
    • 86% of voters who voted Republican were registered Republican.

    What is the probability that a voter voted Republican given that the voter was registered Republican?

    • Worked-Out Solution

      Event 1 is "the voter voted Republican."

      Event 2 is "the voter is a registered Republican."

      Event 2, given event 1 is "the person who voted Republican is a registered Republican."

      A set diagram might help you see what is going on. In the diagram, you can see that 86% of the 53% of the people who voted Republican are registered Republicans. If you are given that the selected voter is a registered Republican, then you have restricted your total population to the 61%. So, given that a person is a registered Republican, the probability that the person voted Republican is

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      ______     ___     _    _     _____    _    _   
     /_   _//   / _ \\  | \  / ||  |  ___|| | |  | || 
       | ||    | / \ || |  \/  ||  | ||__   | |/\| || 
      _| ||    | \_/ || | .  . ||  | ||__   |  /\  || 
     /__//      \___//  |_|\/|_||  |_____|| |_// \_|| 
     `--`       `---`   `-`  `-`   `-----`  `-`   `-` 
                                                      
    
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  • 26. Bayes' Theorem

    For any two events with probabilities greater than 0,

    You have the following information about voters in a local congressional election.

    • 74% of voters were registered Democrat
    • 62% of voters voted Democrat
    • 79% of voters who voted Democrat were registered Democrat

    What is the probability that a voter voted Democrat given that the voter was registered Democrat?

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      ______   _    _    _    _    __   __   __   __  
     /_   _// | || | || | \  / ||  \ \\/ //  \ \\/ // 
       | ||   | || | || |  \/  ||   \ ` //    \   //  
      _| ||   | \\_/ || | .  . ||    | ||     / . \\  
     /__//     \____//  |_|\/|_||    |_||    /_//\_\\ 
     `--`       `---`   `-`  `-`     `-`'    `-`  --` 
                                                      
    
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