Please answer the following four questions and make a note of your answers. Please answer each question before moving on to the next. The answers will follow.
1. Bat and Ball
A bat and ball together cost $1.10. The bat costs $1 more than the ball. How much does the ball cost?
Please write your answer down.
2. Jack, Anne, and George
Jack is looking at Anne, but Anne is looking at George. Jack is married. George is unmarried. Is a married person looking at an unmarried person?
A. Yes
B. No
C. Cannot be determined
Please write your answer down.
3. Larry’s Occupation
Larry is a quiet, fastidious, and highly-organized man who loves to read and to help other people. Which is more likely to be Larry’s profession?
A. Librarian
B. Farmer
Please write your answer down.
4. Four-Card Selection Task
This is a fact about the cards below: each has a letter on one side and a number on the other. Here is a rule about these cards that needs to be tested: If a card has a vowel on one side, it has an even number on its reverse.
Which card or cards must you turn over in order to test whether this rule is true or false?
Please write your answer down.
Please do not read further until you have answered the questions above and recorded your answers.
These questions provide a good test of some of your cognitive habits. For example:
• Whether you are inclined to check your math before finalizing an answer.
• Whether you are easily “framed” by the way a particular question is constructed.
• Whether you search for and consider all the possibilities before settling on an answer.
• Whether you will resist an easy-but-incorrect answer that seems to “make sense” in order to find a more difficult one that not only makes sense, but is also correct.
• Whether you will substitute an easier question for the question that was actually asked.
• Whether you have a good grasp on the basics of logic and probability.
• Whether you are overconfident in your minute-by-minute grasp on the realities of a problem.
• How you handle problems for which you don’t have a “gut feeling” about what the right answer is.
So, How Did You Do?
The questions in this pretest will be discussed throughout this book (along with other questions not shown here), but here’s a brief discussion of each, including the correct answers. If you’re like most, you’ll be disappointed in your results and this book will be a real eye-opening experience for you.
1. Bat and Ball. The ball costs $0.05. The bat is $1.00 more than the ball, which makes that bat $1.05. When you add them together, you get this: $0.05 + $1.05 = $1.10. Roughly 80% of people get this question wrong, with the most popular answer for the price of the ball being $0.10. Indeed, I got this question wrong when I first saw it, and that’s why I’m writing this book! I said the ball was ten cents. Had I checked my math, I’d have realized quickly that this was wrong. Instead, though, I went with the answer that seemed immediately coherrent, and was so confident in my intuitionfor math that I didn’t check. As it turns out, many people have a strong feeling or hunch that the right answer here is “ten cents”. But alas! We are wrong!
2. Jack, Anne, and George. The correct answer is (A) Yes, a married person is looking at an unmarried person. A majority of people get snagged on this question and end up answering a different question altogether. Instead of answering the question “Is a married person looking at an unmarried person?”, they end up answering as if the question were “Can Anne’s marital state be determined?” The fact of the matter is that we do not need to know Anne’s marital status in order to answer this question. Rather, we simply need to simulate both possibilities for Anne’s marital state (there are only two: married and unmarried). In either case, we find that a married person is looking at an unmarried person. That is, if Anne is unmarried, married Jack is looking at her, and if Anne is married, then married Anne is looking at unmarried George. The key to this problem is considering all the possibilities. I missed this one, too! I answered, “C”.
3. Larry’s Occupation. Larry is most likely a farmer. There are a couple of issues with this question that tend to snag people. One is that it requires information not given, and the second is that it tends to evoke a “coherrence bias” in our thinking. Many guess “librarian” because the problem presents a portrait of a “quiet, fastidious, and organized” person who “loves to read and to help other people”, and then offers you a chance to match that portrait with the occupation of librarian. It fits, right? So “librarian” must be the right answer, it seems—but it’s not. There’s more to this question than meets the eye. In order to answer it well, we also need to have some idea about the “base rate” of occupations by sex. In the US, there are currently about 34,000 male librarians, and over 1.8 million male farmers. That’s about 53 times more male farmers than librarians, so it is more probable that any particular male will be a farmer than a librarian. We can further clear up the confusion over the coherrence of the stereotype by asking ourselves whether it is possible that a farmer could be a “quiet, fastidious, and organized man who loves to read and to help other people”. Of course, it is!
4. Four-Card Selection Test. In order to test the rule, you must turn over the A card and the 5 card. Turning the K and 8 are not necessary. Here are discussions on each individual card:
K. Turning the K card does not help because we know that each card has a letter on one side and a number on the other. The rule concerns only those cards with a vowel on the letter side. So turning the K card does not help us test the rule.
A. Turning the A card is necessary because we know it has a number on the reverse, and we can check to see whether that number is even (following the rule) or odd (breaking the rule.)
8. Turning the 8 card is nearly irresistible for many, but it does not test the rule. Most who want to turn it are curious as to whether there is a vowel on the reverse, and if there is not, they will believe that the rule has been violated. The cognitive error in play here, however, is that the real rule (“if vowel, then even”) is being replaced with its reverse (“if even, then vowel”). That is, people assume that if we have an even number on the number side (the 8), then the rule requires a vowel on the opposite side. If-then statements, however, are only intended to tell us something in a left-to-right order; they are not intended to read backwards. In other words, “If A, then B” is not meant to convey also that “If B, then A”. If the 8 card (an even) has a consonant on the reverse, that does not prove or disprove the rule, because the rule is not “If even, then vowel”, but “If vowel, then even.”
5. Turning the 5 card is necessary because it might test the rule. If it has a consonant on its reverse, we learn nothing of value in testing the rule, but if it has a vowel on its reverse, then we would learn that the rule is false, for we are told that any card with a vowel on one face must have an even number on the other.