Five questions to get you thinking
By Kurt Kleiner
Although intelligence as measured by IQ tests is important, so is the ability to think rationally about problems. The surprise is that less intelligent people usually perform just as well as highly intelligent people on problems that test rationality. Here are a few questions that test if you’re a rational thinker.
1. A bat and ball cost $1.10 in total. The bat costs $1 more than the ball. How much does the ball cost?
2. Is the following conclusion logically valid?
Premise 1: All living things need water.
Premise 2: Roses need water.
Therefore, roses are living things.
3. XYZ virus causes a disease in one in every 1,000 people. A test always correctly indicates if a person is infected. The test has a false-positive rate of five per cent – in other words, the test wrongly indicates that the XYZ virus is present in five per cent of the cases in which the person does not have the virus. What is the probability that an individual testing positive actually has the XYZ virus?
4. There are four cards on a table. Each has a letter on one side and a number on the other. The cards look like this:
K A 8 5
Here is a rule: If a card has a vowel on its letter side, it has an even number on its number side. Which card(s) must be turned over to find out if the rule is true or false?
5. According to a comprehensive study by the U.S. Department of Transportation, a particular German car is eight times more likely than a typical family car to kill the occupants of another car in a crash. The U.S. Department of Transportation is considering recommending a ban on the sale of this German car. Do you think the United States should ban the sale of this car?
Answers
1. Five cents. Many people, including students at MIT, Princeton and Harvard, automatically answer 10 cents. After all, a dollar plus 10 cents equals $1.10. But that cognitive shortcut doesn’t work, since it would mean the bat costs only 90 cents more than the ball.
2. No, it is not logical, even though 70 per cent of university students given the problem think it is. Although the conclusion is true, it doesn’t follow from the premises. Consider the same problem worded in a different way:
Premise 1: All insects need oxygen.
Premise 2: Mice need oxygen.
Therefore, mice are insects.
In the original problem, the tendency is to be a cognitive miser, and let the obvious truth of the conclusion substitute for reasoning about its logical validity. (In the second problem, though, our cognitive miser makes the problem easy.)
3. Two per cent. (Most people say 95 per cent.) If one in 1,000 people has the disease, 999 don’t. But with a five per cent false-positive rate, the test will show that almost 50 of them are infected. Of 51 patients testing positive, only one will actually be infected. The math here isn’t especially hard. But thinking the problem through is tricky.
4. A and 5. Ninety per cent of people get this one wrong, usually by picking A and 8. They think they need to confirm the rule by looking for a vowel on the other side of the 8. But the rule only says that vowels must have even numbers, not that consonants can’t. An odd number on the back of the A, or a vowel on the back of the 5, would show that the rule is false.
5. OK, there’s no right or wrong answer here. However, 78 per cent of the people Stanovich sampled thought the German car should be banned. But when he turned the question around so that Germany was considering banning an American car (he was quizzing people in the U.S., by the way), only 51 per cent thought Germany should ban the car. This is an example of “myside bias” – evaluating a problem from a standpoint that is biased toward your own situation.
By Kurt Kleiner
Although intelligence as measured by IQ tests is important, so is the ability to think rationally about problems. The surprise is that less intelligent people usually perform just as well as highly intelligent people on problems that test rationality. Here are a few questions that test if you’re a rational thinker.
1. A bat and ball cost $1.10 in total. The bat costs $1 more than the ball. How much does the ball cost?
2. Is the following conclusion logically valid?
Premise 1: All living things need water.
Premise 2: Roses need water.
Therefore, roses are living things.
3. XYZ virus causes a disease in one in every 1,000 people. A test always correctly indicates if a person is infected. The test has a false-positive rate of five per cent – in other words, the test wrongly indicates that the XYZ virus is present in five per cent of the cases in which the person does not have the virus. What is the probability that an individual testing positive actually has the XYZ virus?
4. There are four cards on a table. Each has a letter on one side and a number on the other. The cards look like this:
K A 8 5
Here is a rule: If a card has a vowel on its letter side, it has an even number on its number side. Which card(s) must be turned over to find out if the rule is true or false?
5. According to a comprehensive study by the U.S. Department of Transportation, a particular German car is eight times more likely than a typical family car to kill the occupants of another car in a crash. The U.S. Department of Transportation is considering recommending a ban on the sale of this German car. Do you think the United States should ban the sale of this car?
Answers
1. Five cents. Many people, including students at MIT, Princeton and Harvard, automatically answer 10 cents. After all, a dollar plus 10 cents equals $1.10. But that cognitive shortcut doesn’t work, since it would mean the bat costs only 90 cents more than the ball.
2. No, it is not logical, even though 70 per cent of university students given the problem think it is. Although the conclusion is true, it doesn’t follow from the premises. Consider the same problem worded in a different way:
Premise 1: All insects need oxygen.
Premise 2: Mice need oxygen.
Therefore, mice are insects.
In the original problem, the tendency is to be a cognitive miser, and let the obvious truth of the conclusion substitute for reasoning about its logical validity. (In the second problem, though, our cognitive miser makes the problem easy.)
3. Two per cent. (Most people say 95 per cent.) If one in 1,000 people has the disease, 999 don’t. But with a five per cent false-positive rate, the test will show that almost 50 of them are infected. Of 51 patients testing positive, only one will actually be infected. The math here isn’t especially hard. But thinking the problem through is tricky.
4. A and 5. Ninety per cent of people get this one wrong, usually by picking A and 8. They think they need to confirm the rule by looking for a vowel on the other side of the 8. But the rule only says that vowels must have even numbers, not that consonants can’t. An odd number on the back of the A, or a vowel on the back of the 5, would show that the rule is false.
5. OK, there’s no right or wrong answer here. However, 78 per cent of the people Stanovich sampled thought the German car should be banned. But when he turned the question around so that Germany was considering banning an American car (he was quizzing people in the U.S., by the way), only 51 per cent thought Germany should ban the car. This is an example of “myside bias” – evaluating a problem from a standpoint that is biased toward your own situation.
