This article in Scientific American, by Keith Stanovich, makes some solid points about the limits of IQ.
No doubt you know several folks with perfectly respectable IQs who repeatedly make poor decisions. The behavior of such people tells us that we are missing something important by treating intelligence as if it encompassed all cognitive abilities.
I highly recommend going over to SA and reading the whole article. But just for fun, here are some “brain teasers” that Mr. Stanovich included in his article, along with his answers, given in the footnotes.
1. Jack is looking at Anne, but Anne is looking at George. Jack is married, but George is not. Is a married person looking at an unmarried person?
C) Cannot be determined
2. A bat and a ball cost $1.10 in total. The bat costs $1 more than the ball. How much does the ball cost?
3. I’m going to skip question three, because although it illustrates an interesting and important finding about how partisanship kills thoughtfulness, it’s not fun as a stand-alone brain-teaser. But you can read it at Scientific American.3
4. Imagine that XYZ viral syndrome is a serious condition that affects one person in 1,000. Imagine also that the test to diagnose the disease always indicates correctly that a person who has the XYZ virus actually has it. Finally, suppose that this test occasionally misidentifies a healthy individual as having XYZ. The test has a false-positive rate of 5 percent, meaning that the test wrongly indicates that the XYZ virus is present in 5 percent of the cases where the person does not have the virus.
Next we choose a person at random and administer the test, and the person tests positive for XYZ syndrome. Assuming we know nothing else about that individual’s medical history, what is the probability (expressed as a percentage ranging from zero to 100) that the individual really has XYZ?
5. An experiment is conducted to test the efficacy of a new medical treatment. Picture a 2 x 2 matrix that summarizes the results as follows:
Improvement No Improvement
Treatment Given 200 75
No Treatment Given 50 15
As you can see, 200 patients were given the experimental treatment and improved; 75 were given the treatment and did not improve; 50 were not given the treatment and improved; and 15 were not given the treatment and did not improve. Answer this question with a yes or no: Was the treatment effective?
6. As seen in the diagram, four cards are sitting on a table. Each card has a letter on one side and a number on the other. Two cards are letter-side up, and two of the cards are number-side up. The rule to be tested is this: for these four cards, if a card has a vowel on its letter side, it has an even number on its number side. Your task is to decide which card or cards must be turned over to find out whether the rule is true or false. Indicate which cards must be turned over.
- More than 80 percent of people choose C. But the correct answer is A. Here is how to think it through logically: Anne is the only person whose marital status is unknown. You need to consider both possibilities, either married or unmarried, to determine whether you have enough information to draw a conclusion. If Anne is married, the answer is A: she would be the married person who is looking at an unmarried person (George). If Anne is not married, the answer is still A: in this case, Jack is the married person, and he is looking at Anne, the unmarried person. [↩]
- Many people give the first response that comes to mind—10 cents. But if they thought a little harder, they would realize that this cannot be right: the bat would then have to cost $1.10, for a total of $1.20. [↩]
- Filler footnote. [↩]
- The most common answer is 95 percent. But that is wrong. People tend to ignore the first part of the setup, which states that only one person in 1,000 will actually have XYZ syndrome. If the other 999 (who do not have the disease) are tested, the 5 percent false-positive rate means that approximately 50 of them (0.05 times 999) will be told they have XYZ. Thus, for every 51 patients who test positive for XYZ, only one will actually have it. Because of the relatively low base rate of the disease and the relatively high false-positive rate, most people who test positive for XYZ syndrome will not have it. The answer to the question, then, is that the probability a person who tests positive for XYZ syndrome actually has it is one in 51, or approximately 2 percent. [↩]
- Most people will say yes. They focus on the large number of patients (200) in whom treatment led to improvement and on the fact that of those who received treatment, more patients improved (200) than failed to improve (75). Because the probability of improvement (200 out of 275 treated, or 200/275 = 0.727) seems high, people tend to believe the treatment works. But this reflects an error in scientific thinking: an inability to consider the control group, something that (disturbingly) even physicians are often guilty of. In the control group, improvement occurred even when the treatment was not given. The probability of improvement with no treatment (50 out of 65 not treated, or 50/65 = 0.769) is even higher than the probability of improvement with treatment, meaning that the treatment being tested can be judged to be completely ineffective. [↩]
- Most people get the answer wrong, and it has been devilishly hard to figure out why. About half of them say you should pick A and 8: a vowel to see if there is an even number on its reverse side and an even number to see if there is a vowel on its reverse. Another 20 percent choose to turn over the A card only, and another 20 percent turn over other incorrect combinations. That means that 90 percent of people get it wrong.
Let’s see where people tend to run into trouble. They are okay with the letter cards: most people correctly choose A. The difficulty is in the number cards: most people mistakenly choose 8. Why is it wrong to choose 8? Read the rule again: it says that a vowel must have an even number on the back, but it says nothing about whether an even number must have a vowel on the back or what kind of number a consonant must have. (It is because the rule says nothing about consonants, by the way, that there is no need to see what is on the back of the K.) So finding a consonant on the back of the 8 would say nothing about whether the rule is true or false. In contrast, the 5 card, which most people do not choose, is essential. The 5 card might have a vowel on the back. And if it does, the rule would be shown to be false because that would mean that not all vowels have even numbers on the back. In short, to show that the rule is not false, the 5 card must be turned over. [↩]