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8.3 Expected Value

8.3 Expected Value
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  • 11. Comparing Options

    Compare the two options. (See Example 5.)

    • Worked-Out Solution

      According to expected value, the second option would pay better in the long run. Which would you choose? Would you choose a 100% chance of getting $1000 or a 60% chance of getting $2000?

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      ______   __   __   _____       ___      ______  
     /_   _//  \ \\/ // |  __ \\    / _ \\   /_   _// 
     `-| |,-    \ ` //  | |  \ ||  | / \ ||  `-| |,-  
       | ||      | ||   | |__/ ||  | \_/ ||    | ||   
       |_||      |_||   |_____//    \___//     |_||   
       `-`'      `-`'    -----`     `---`      `-`'   
                                                      
    
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  • 12. Comparing Options

    Compare the two options. (See Example 5.)

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      ____     _    _   __    __   __   __   _    _   
     |  _ \\  | || | || \ \\ / //  \ \\/ // | || | || 
     | |_| || | || | ||  \ \/ //    \ ` //  | || | || 
     | .  //  | \\_/ ||   \  //      | ||   | \\_/ || 
     |_|\_\\   \____//     \//       |_||    \____//  
     `-` --`    `---`       `        `-`'     `---`   
                                                      
    
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  • 13. Comparing Options

    Compare the two options. (See Example 5.)

    • Worked-Out Solution

      According to expected value, the second option would pay better in the long run. Remember, however, that 10% of the time, the second option results in $0.

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      ____       ___     _____      _____     _  _   
     |  _ \\    / _ \\  |  __ \\   |  ___||  | \| || 
     | |_| ||  / //\ \\ | |  \ ||  | ||__    |  ' || 
     | .  //  |  ___  ||| |__/ ||  | ||__    | .  || 
     |_|\_\\  |_||  |_|||_____//   |_____||  |_|\_|| 
     `-` --`  `-`   `-`  -----`    `-----`   `-` -`  
                                                     
    
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  • 14. Comparing Options

    Compare the two options. (See Example 5.)

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     _    _     ______    ______   __   __   ______   
    | \  / ||  /_   _//  /_   _//  \ \\/ // |      \\ 
    |  \/  ||   -| ||-     | ||     \ ` //  |  --  // 
    | .  . ||   _| ||_    _| ||      | ||   |  --  \\ 
    |_|\/|_||  /_____//  /__//       |_||   |______// 
    `-`  `-`   `-----`   `--`        `-`'   `------`  
                                                      
    
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  • 15. Investment Comparison

    You want to invest $1000. Find the expected values for the two investments. (See Example 6.)

    • Worked-Out Solution

      A spreadsheet helps to organize the calculations.

        Speculative Investment

        Conservative Investment

      The speculative investment has a much greater expected value. By choosing this investment, however, you also have a better chance of losing everything.

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     _____      _____    _    _      ___      _  __  
    |  __ \\   |  ___|| | || | ||   / _ \\   | |/ // 
    | |  \ ||  | ||__   | || | ||  | / \ ||  | ' //  
    | |__/ ||  | ||__   | \\_/ ||  | \_/ ||  | . \\  
    |_____//   |_____||  \____//    \___//   |_|\_\\ 
     -----`    `-----`    `---`     `---`    `-` --` 
                                                     
    
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  • 16. Investment Comparison

    You want to invest $1000. Find the expected values for the two investments. (See Example 6.)

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      ______    ______   __   _      ___              
     /_   _//  /_   _// | || | ||   / _ \\    ____    
       | ||     -| ||-  | '--' ||  / //\ \\  |    \\  
      _| ||     _| ||_  | .--. || |  ___  || | [] ||  
     /__//     /_____// |_|| |_|| |_||  |_|| |  __//  
     `--`      `-----`  `-`  `-`  `-`   `-`  |_|`-`   
                                             `-`      
    
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