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8.2 Estimating Likelihood

8.2 Estimating Likelihood
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  • 23. Comparing Probabilities

    You randomly choose a card from a standard deck of playing cards. Find the theoretical probability of choosing a card of each suit.

    • Worked-Out Solution

      There are 52 cards in the deck and 13 of each suit. So, the probability of drawing any particular suit is

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      _  _    _    _     _  __    _____    _    _   
     | \| || | || | ||  | |/ //  |  ___|| | |  | || 
     |  ' || | || | ||  | ' //   | ||__   | |/\| || 
     | .  || | \\_/ ||  | . \\   | ||__   |  /\  || 
     |_|\_||  \____//   |_|\_\\  |_____|| |_// \_|| 
     `-` -`    `---`    `-` --`  `-----`  `-`   `-` 
                                                    
    
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  • 24. Comparing Probabilities

    You have a standard deck of cards. The bar graph shows the results of randomly choosing 1 card, recording its suit, and placing it back in the deck for 50 trials. Find the experimental probability of choosing a card of each suit.

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      ______   __   __   _    _    __   __   ______   
     /_____//  \ \\/ // | || | ||  \ \\/ // |      \\ 
     `____ `    \ ` //  | || | ||   \ ` //  |  --  // 
     /___//      | ||   | \\_/ ||    | ||   |  --  \\ 
     `__ `       |_||    \____//     |_||   |______// 
     /_//        `-`'     `---`      `-`'   `------`  
     `-`                                              
    
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  • 25. Comparing Probabilities

    Compare the probabilities found in Exercises 23 and 24.

    • Worked-Out Solution

      The theoretical and experimental probabilities for drawing each suit are as follows.

      Anyone who has played cards recognizes that a theoretical probability is the result you expect over hundreds or even thousands of card hands. For a small number of hands (like 50), the results can vary. This is the whole point of card games, isn't it? You are hoping that you will have good "luck" and that your "experimental probability" will be better than the average.

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      ____       ___      ______     ___      ______  
     |  _ \\    / _ \\   /_____//   / _ \\   /_   _// 
     | |_| ||  | / \ ||  `____ `   | / \ ||    | ||   
     | .  //   | \_/ ||  /___//    | \_/ ||   _| ||   
     |_|\_\\    \___//   `__ `      \___//   /__//    
     `-` --`    `---`    /_//       `---`    `--`     
                         `-`                          
    
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  • 26. Addition Rule

    The probability that one of two events occurs is

    You randomly choose a card from a standard deck of cards. Find the probability of choosing a heart or a 6.

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      _____     _____    _    _      ___     _    _   
     |__  //   |  ___|| | || | ||   / _ \\  | \  / || 
       / //    | ||__   | || | ||  | / \ || |  \/  || 
      / //__   | ||__   | \\_/ ||  | \_/ || | .  . || 
     /_____||  |_____||  \____//    \___//  |_|\/|_|| 
     `-----`   `-----`    `---`     `---`   `-`  `-`  
                                                      
    
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  • 27. Addition Rule

    The probability that one of two events occurs is

    You randomly choose a card from a standard deck of cards. Find the probability that the card is black or a 2.

    • Worked-Out Solution

      Here are the individual probabilities.

      So the probability of drawing a black card or a 2 is about

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      _____     _____    ______     _____    _    _   
     |__  //   |  ___|| |      \\  |  ___|| | || | || 
       / //    | ||__   |  --  //  | ||__   | || | || 
      / //__   | ||__   |  --  \\  | ||__   | \\_/ || 
     /_____||  |_____|| |______//  |_____||  \____//  
     `-----`   `-----`  `------`   `-----`    `---`   
                                                      
    
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  • 28. Addition Rule

    The probability that one of two events occurs is

    You randomly choose a card from a standard deck of cards. Find the probability of choosing a face card or a diamond.

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