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8.4 Expecting the Unexpected

8.4 Expecting the Unexpected
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  • 9. Rolling Dice

    You roll three dice. What is the probability of rolling at least two of the same number? (See Example 3 and Example 4.)

    • Worked-Out Solution

      The probability that all three numbers are different is

      So, the probability that at least two numbers are the same is

      In other words, if you rolled 3 dice 1000 times, you would expect at least two numbers to be alike around 444 times.

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      ______     ___                 ___      _  __  
     /_   _//   / _ \\      ___     / _ \\   | |/ // 
       | ||    / //\ \\    /   ||  / //\ \\  | ' //  
      _| ||   |  ___  ||  | [] || |  ___  || | . \\  
     /__//    |_||  |_||   \__ || |_||  |_|| |_|\_\\ 
     `--`     `-`   `-`     -|_|| `-`   `-`  `-` --` 
                             `-`                     
    
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  • 10. Rolling Dice

    You roll five dice. What is the probability of rolling at least two of the same number? (See Example 3 and Example 4.)

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      ______   _    _              _    _    __   _   
     /_____// | || | ||   ____    | || | || | || | || 
     `____ `  | || | ||  |    \\  | || | || | '--' || 
     /___//   | \\_/ ||  | [] ||  | \\_/ || | .--. || 
     `__ `     \____//   |  __//   \____//  |_|| |_|| 
     /_//       `---`    |_|`-`     `---`   `-`  `-`  
     `-`                 `-`                          
    
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  • 11. Rolling Dice

    You roll six dice. What is the probability of rolling at least one odd number? (See Example 3 and Example 4.)

    • Worked-Out Solution

      The probability that all six numbers are even is

      So, the probability that at least one number is odd is

      In other words, if you rolled 6 dice 1000 times, you would expect at least one number to be odd about 984 times.

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      _  _      ___     _____     __   __   _    _   
     | \| ||   / _ \\  |  __ \\   \ \\/ // | || | || 
     |  ' ||  | / \ || | |  \ ||   \ ` //  | || | || 
     | .  ||  | \_/ || | |__/ ||    | ||   | \\_/ || 
     |_|\_||   \___//  |_____//     |_||    \____//  
     `-` -`    `---`    -----`      `-`'     `---`   
                                                     
    
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  • 12. Rolling Dice

    You roll seven dice. What is the probability of rolling at least two of the same number? (See Example 3 and Example 4.)

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     _    _      ___      _____      ___      _____   
    | \  / ||   / _ \\   |__  //    / _ \\   / ____|| 
    |  \/  ||  / //\ \\    / //    / //\ \\ / //---`' 
    | .  . || |  ___  ||  / //__  |  ___  ||\ \\___   
    |_|\/|_|| |_||  |_|| /_____|| |_||  |_|| \_____|| 
    `-`  `-`  `-`   `-`  `-----`  `-`   `-`   `----`  
                                                      
    
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  • 13. Drawing Marbles from a Bag

    There are 26 marbles in a bag. The graph shows the color distribution of the marbles. Assume that you randomly draw one marble from the bag and you put the marble back before drawing another.

    You randomly draw three marbles from the bag. What is the probability that none of the marbles are red? (See Example 3 and Example 4.)

    • Worked-Out Solution

      The probability that none of the marbles are red is

      In other words, if you drew 3 marbles from the bag 1000 times, you would expect none of the marbles to be red about 455 times.

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    __    __    ______  __    __   __   __   _    _   
    \ \\ / //  /_   _// \ \\ / //  \ \\/ // | \  / || 
     \ \/ //    -| ||-   \ \/ //    \ ` //  |  \/  || 
      \  //     _| ||_    \  //      | ||   | .  . || 
       \//     /_____//    \//       |_||   |_|\/|_|| 
        `      `-----`      `        `-`'   `-`  `-`  
                                                      
    
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  • 14. Drawing Marbles from a Bag

    There are 26 marbles in a bag. The graph shows the color distribution of the marbles. Assume that you randomly draw one marble from the bag and you put the marble back before drawing another.

    You randomly draw five marbles from the bag. What is the probability that none of the marbles are yellow? (See Example 3 and Example 4.)

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      _  __     ___      _  __     ___      ____    
     | |/ //   / _ \\   | |/ //   / _ \\   |  _ \\  
     | ' //   | / \ ||  | ' //   | / \ ||  | |_| || 
     | . \\   | \_/ ||  | . \\   | \_/ ||  | .  //  
     |_|\_\\   \___//   |_|\_\\   \___//   |_|\_\\  
     `-` --`   `---`    `-` --`   `---`    `-` --`  
                                                    
    
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  • 15. Drawing Marbles from a Bag

    There are 26 marbles in a bag. The graph shows the color distribution of the marbles. Assume that you randomly draw one marble from the bag and you put the marble back before drawing another.

    You randomly draw four marbles from the bag. What is the probability that at least one of the marbles is green? (See Example 3 and Example 4.)

    • Worked-Out Solution

      The probability that none of the marbles are green is

      So, the probability that at least one of the marbles is green is

      In other words, if you drew 4 marbles from the bag 1000 times, you would expect at least one of the marbles to be green about 770 times.

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      ______     ___     __   _    __   __    ____    
     /_____//   / _ \\  | || | ||  \ \\/ //  |  _ \\  
     `____ `   / //\ \\ | '--' ||   \ ` //   | |_| || 
     /___//   |  ___  ||| .--. ||    | ||    | .  //  
     `__ `    |_||  |_|||_|| |_||    |_||    |_|\_\\  
     /_//     `-`   `-` `-`  `-`     `-`'    `-` --`  
     `-`                                              
    
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  • 16. Drawing Marbles from a Bag

    There are 26 marbles in a bag. The graph shows the color distribution of the marbles. Assume that you randomly draw one marble from the bag and you put the marble back before drawing another.

    You randomly draw six marbles from the bag. What is the probability that at least one marble is blue? (See Example 3 and Example 4.)

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