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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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