Ron Larson
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8.4 Expecting the Unexpected

8.4 Expecting the Unexpected
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  • 17. Coin Conundrum

    Your friend flips two coins and tells you that at least one coin landed heads up. What is the probability that both coins landed heads up? (See Example 5.)

    • Worked-Out Solution

      When two coins are flipped, there are four possible outcomes.

      By your friend telling you that at least one coin landed heads up, you know that the outcome is one of the first three.

      Because each of the three outcomes (with at least one head) is equally likely, the probability that both coins landed heads up is

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      ____       ___      ____      _____     ______  
 |  _ \\    / _ \\   |  _ \\   |  ___||  /_____// 
 | |_| ||  / //\ \\  | |_| ||  | ||__    `____ `  
 | .  //  |  ___  || | .  //   | ||__    /___//   
 |_|\_\\  |_||  |_|| |_|\_\\   |_____||  `__ `    
 `-` --`  `-`   `-`  `-` --`   `-----`   /_//     
                                         `-`
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  • 18. Boy or Girl Paradox

    Your friend has two children. At least one of the children is a boy. What is the probability that the other child is a girl? (See Example 5.)

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     ______      ___                ______   __   _   
|      \\   / _ \\    ____     /_   _// | || | || 
|  --  //  | / \ ||  |    \\    -| ||-  | '--' || 
|  --  \\  | \_/ ||  | [] ||    _| ||_  | .--. || 
|______//   \___//   |  __//   /_____// |_|| |_|| 
`------`    `---`    |_|`-`    `-----`  `-`  `-`  
                     `-`
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  • 19. Treasure Beyond Measure

    You are a contestant on the fictional game show Treasure Beyond Measure. The host presents you with five unopened treasure chests. One chest contains $1,000,000. The other four contain nothing. You randomly choose two of the chests. The host, knowing which chest contains the money, opens two of the remaining chests and shows that they contain nothing. The host then asks you, "Do you want to switch your two chests for the one chest that I didn't open?" Based on probability, what should you do? (See Example 5.)

    • Worked-Out Solution

      You can answer this question by following the model given in Example 5. Because it is irrelevant, assume that you choose Chests 1 and 2, and the host chooses two of the remaining chests that have nothing.

      By staying, you will win 2 times out of 5. By switching, you will win 3 times out of 5. So, you should switch.

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       _____     ___      ______   __   __   _____    
  / ___//   / _ \\   /_____//  \ \\/ // |  __ \\  
  \___ \\  | / \ ||  `____ `    \ ` //  | |  \ || 
  /    //  | \_/ ||  /___//      | ||   | |__/ || 
 /____//    \___//   `__ `       |_||   |_____//  
`-----`     `---`    /_//        `-`'    -----`   
                     `-`
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  • 20. Ultimate Monty Hall

    You are a contestant on the fictional game show Ultimate Monty Hall. The host presents you with 101 unopened doors. Behind one door is a car. Behind the other 100 doors are goats. You randomly choose 50 of the doors. The host, knowing the location of the car, reveals the goats behind 50 of the remaining doors. The host then asks you, "Do you want to switch your 50 doors for the 1 door that I didn't open?" Based on probability, what should you do? (See Example 5.)

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     _____     __   __             _    _     _____   
|  __ \\   \ \\/ //     ___   | || | ||  /  ___|| 
| |  \ ||   \ ` //     /   || | || | || | // __   
| |__/ ||    | ||     | [] || | \\_/ || | \\_\ || 
|_____//     |_||      \__ ||  \____//   \____//  
 -----`      `-`'       -|_||   `---`     `---`   
                         `-`
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  • 21. Card Puzzle

    You deal two of the cards shown to your friend. Your friend says, "I have at least one ace." What is the probability that your friend's other card is an ace? (See Example 5 and Example 6.)

    • Worked-Out Solution

      There are 6 possible ways to deal 2 cards from 4 cards.

      Your friend is telling you that he has one of the five possible hands that have at least one ace. Because each of these five hands is equally likely, the probability that your friend has two aces is

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      _  __   _    _     _  __     ___     __   _   
 | |/ // | || | ||  | |/ //   / _ \\  | || | || 
 | ' //  | || | ||  | ' //   | / \ || | '--' || 
 | . \\  | \\_/ ||  | . \\   | \_/ || | .--. || 
 |_|\_\\  \____//   |_|\_\\   \___//  |_|| |_|| 
 `-` --`   `---`    `-` --`   `---`   `-`  `-`
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  • 22. Card Puzzle

    You deal two of the cards shown to your friend. Your friend says, "I have the ace of spades." What is the probability that your friend's other card is an ace? (See Example 5 and Example 6.)

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