Five questions to get you thinking

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.

## Reader Comments

Disregarding question #5, I got them right, except #4 (I said “A” only). The thing is, if I encountered these questions in normal life, I would have given all wrong answers. But because I’m aware that this is a test, and the answer is not the most apparent one, I spent more time analyzing the problems.

So I wonder: do “rational” people consider every problem a “tricky” one? Or are they just like me, except that their thought process changes depending on how they perceive the issue?

It would be great to know if I could trick my brain to think “rationally” all of the time, rather than only when I’m answering questions designed to test my rationality.

On question #3, you would be correct if your question was, ‘Does he have the disease?’

But you asked if he had the virus, in which case the answer is 95%

I agree with AE, above. I used to play a trivial pursuit game in a pub. We usually won, but we got some wrong because the answer was too easy. We assumed there must be a trick. But there wasn’t.

Has anyone read Dan Ariely’s book. “Predictably Irrational?”

Regarding question four, wouldn’t turning over the K and 8 also confirm the rule?

K is not a vowel, so turning it over won’t confirm the rule. If 8 has a consonant on the back, it won’t confirm the rule (since the rule says only that vowels must even numbers on the back; it doesn’t say that consonants can’t).

Regarding question #4, The data seems insufficient to actually

verifythe rule. Turning over the 5 and finding a consonant only proves that there can be consonants on the back of odd numbered cards.I suggest that question #5 is unanswerable.

One factor is that the German car could be extremely heavy because it has the capacity to transport many people or has bullet-proofed body for security. Thus its propensity to kill people in accidents is simply caused by its mass.

So unless the government wants to place limits on the allowable mass of vehicles, it makes no sense to ban a specific make and model.

Question #4 is deliberately misleading and short on information. You can’t prove whether the rule is true, only if it is false, unless you state that the rule only applies to these 4 cards.

Question #3 is also bogus, because they are not asking about the disease, but the infection with XYZ virus.

The correct answer to #3 is actually slightly below 2%. It’s 1 in 51, not 1 in 50. For those who can’t figure it out, here’s another explanation. Suppose you have 1000 people tested. 5% of them, on average, will have a false positive. That is 50 people. One of them will have a true positive, and actually have the virus. So you have 51 positive results, only one of which actually has the virus. Therefore the odds that any given individual with a positive result has the virus is 1 in 51.

Re question #4, why are two cards needed to prove the point? If A is turned and the back is an even number, the premise is correct. If the number is uneven, the premise is wrong.

With respect to question #2, I think the answer is logically correct although it may not be factually correct

I have to say that the question about the virus is worded incorrectly for the outcome. I considered the answer given but I had disregarded that since the contraction of the XYZ virus is what we are asked to determine, NOT the disease itself. Try to think of it as HIV. Just because someone has HIV, does not mean they have AIDS. They still have HIV. The last sentence should be worded, “What is the probability that an individual testing positive actually has the disease caused by the XYZ virus?”

I love the article and quiz. This author has made some very important points about what needs to be taught in our schools. As an educator, I am constantly faced with the lack of rational thought by adults and children alike. I’m very interested in learning methods of teaching students how to think rationally. Interestingly, I have indeed known individuals with very low IQs who seem to think more rationally than some who are highly educated.

On Question #3, I agree with Yohonna, who provided a great analogy. The answer, given the way the question is stated, is actually slightly larger than 95%. Suppose 200 people are tested, of which 100 have the virus and 100 don’t. Since the test always correctly identifies those that are infected, the 100 who have the virus will test positive. Of the 100 who don’t have the virus, five will test positive. Therefore, 105 people in total will test positive, of which five actually don’t have the virus. The probability of having the virus if you test positive is therefore 100/105 or 95.24%.

I agree with Bruce. I would want to buy that German car but my politicians would ban it, so I couldn’t.

In Q4, the question should be, ‘How many cards need to be inspected to disprove the rule. In what order would you turn over the cards to keep the number turned to a minimum?’

The first sentence of #3 is actually ambiguous. “XYZ virus causes a disease in one in every 1,000 people” is effectively elliptical, it could be short for “in 1 in every 1000 people who have the virus”, or “in 1 in every 1000 people in the population” where having the virus *always* entails having the disease. Obviously the author intended the latter and was probably unaware of the other interpretation, which I suspect would not have occurred to so many people before HIV/AIDS brought the importance of this distinction to the fore.

Re JB’s comment on #4, the only sensible interpretation of “find out if the rule is true or false” is “determine whether the statement is true of (all) the cards on the table”. Considering other cards that might exist in the world makes the problem unanswerable and uninteresting. Once we agree on that, there is no issue of confirming vs. disconfirming, it makes no difference in what order you turn them over, and the question already did ask for the minimum number of cards to turn because it said ‘how many *must* be turned over’; the proffered solution is correct.

It would be nice if the author had cited sources for these examples. #4 is in every cognitive psych textbook and comes from work by Wason ca. 1970.

As noted by post-ers #2 and #9, Question 3’s first sentence refers to a disease that is never mentioned later, with predictably misleading results.

The sentence seems to mean that the virus causes disease in only 0.1% of those that are infected; but if so, it provides no information about the incidence of infection in the population at large.

If the first sentence is instead intended to mean “0.1% of the population is infected”, then the ratio of infected:positive in a sample of that population is indeed 1:51, as pointed out by post-er #10.

Post-er #13’s analysis supposes a 1:1 ratio of infected:uninfected, an unwarranted assumption.

With regards to #4 and why the 5 has to be turned: “If a card has a vowel on its letter side, it has an even number on its number side” implies “if it has an uneven number on its number side, it does not have a vowel on its letter side,” which is what must be tested.

I got the answers to questions 1, 2 and 5, while experiencing the same “interpretive” confusion as others with question 3, but with a variant. The question could benefit from being clearer.

Here’s where I got hung up with the wording. I calculated the rate of disease at 1/1000, or, to make the math easier, 100 out of every 100,000 people. Of those 100,000 people, a false positive would also be registered at a rate of 5 people along with every 100 for which a “true positive” was properly recorded. By this interpretation, it would mean that in total, 105 people will test positive out of 100,000 people, with 100 of those 100,000 being true positives. So a true positive in this reckoning would be every 100 out of every 105 people divided by 1,000, or roughly 0.095 percent of the population being tested as positive and actually having the “virus.” This is another way of looking at the “probability that an individual testing positive actually has the XYZ virus” among the general population.

Question 2 demonstrates a standard logical fallacy. When you say, “if A, then B, and if B, then C” then it is logical to say “therefore, if A, then C.” But you can’t say: “If A then C,” “if B then C,” therefore “since we know C is true, A must equal B.”

Regarding question 4, I was probably overthinking it and concur with others who observe that two cards don’t really prove any “rule”, unless the total universe of cards for which the rule is supposed to apply is just the four cards. Without specifying what the “card universe” is that the “rule” should apply to, one might argue that no one card or two cards can “prove” the rule, they can only prove that the rule has not yet been disproven.

With respect to question 5, may I suggest another way of looking at the problem. Just because a particular brand of car is involved in more fatal accidents than a “typical” car, it does not necessarily mean it’s unsafe. It could be an extremely popular car, meaning that there are many more of them on the road. Let’s say, for example, that of the nine cars in the “typical family car” category, which includes the VW Jetta, each of the other cars has a 6.25% share of the market, and the Jetta has 50%. This would mean that for each “typical” competitor on the road, there would be eight Jettas. So while it might appear that the car is “eight times more likely to kill” occupants of other cars, this is a perfectly “typical” outcome given the circumstances.

Regarding question 3, shouldn’t it be slightly higher than 1/51? i.e. 1/50.95

100 in 100,000 have the virus, and will also get that result on the test.

Meaning 99,900 don’t have it!

“Five per cent of the cases in which the person does not have the virus” means 99900*0.05=4995 would get a test-result saying they got the virus but they don’t.

So test-results with a positive answer would thus be 4995+100=5095 people.

Only 100 of those got the virus, so it would be slightly less than 2% (100/5095=1/50.95=~1.96%).

I answered:

1. 5 cents

2. No.

3. I answered 1 out of 51. I consider that to be correct, as if 5 percent out of a 1000 gives 50 false positives, and that there is always 1 infected person in a 100 without exception, it would mean that 50 false positives plus 1 positive would show positive on the tests. Out of 51 the chances you have the disease is 1:51.

4. A and 5.

If there is a vowel behind 5, the rule is broken. If there is an odd number behind A, the rule is broken.

K A 8 5

Turn around K, either number can be behind it, and no rule is broken.

Turn around A, if there is an odd number behind it, the rule is broken.

Turn around 8, either a vowel or a consonant can be behind the number, and no rules are broken, because the rule is, that vowels must have an even number on the otherside, but an even number doesn’t have to have a vowel on the other side.

Turn around 5, if there is a vowel behind this card, the rule is broken, as the vowel should have an even number on the other side, not an odd number.

Simple, eh?

5. What is the survivability rate inside both cars? If all people had the german cars, then more people would presumably die, because none mentioned the chance of the people in the car surviving. That’s how I thought about it. If the survivability inside neither car is mentioned, it is safe to assume it is the same. In real life you could study it of course, but in a hypothetical question like this one can’t.

If you drive an american car and you crash, the chances you kill someone else are, lets say, 1:100. The chances you die depend solely on what car the other driver is driving.

If you drive a german car, the chances you kill someone else would then be 8:100. The chances you die depend solely on what car the other driver is driving.

If 2 american cars crash, there is a 1:100 chance both drivers will die.

If 2 of the german cars crash, there is an 8:100 chance that both the drivers will die.

If an american car and a german car crash, the american has an 8:100 chance to die, the german has a 1:100 chance to die (Here germans drive german cars and americans drive american cars).

If everyone were to drive the american cars, it would mean that there was always a 1:100 chance for you to die in a crash with another vehicle.

If everyone were to drive the german cars, one would have an 8:100 chance of dying in a car crash.

So, if thought of in this way, the more german cars there are in any possible scenario, the more people will eventually end up dying, because we can’t assume that being inside a german car would be any safer.

I think there is enough data here to ban the car, if one thought these increased odds would be enough to ban a car, but of course I might be, and most likely am wrong and making false assumptions in some way. I just don’t see it at the moment. I shouldn’t hold a biased viewpoint on this, because I don’t live in either country, and I can only hope my rationality excludes bias.

answer to question #3

They (the doctors/scientists) take the assumption “1 in 1,000 people have the virus” based on that test which have 5% false positive rates. Right?

Let’s take 100,000 people, the test will show 100 of them have the virus. But since the test have 5% false positive rates, so 5% of that 100 people actually don’t have the virus, so people who actually have the virus is 95 people out of 100.000 = 0.00095 = 0.095% = slightly less than 1%.

Btw, I got the other questions correct, except #5 which I answered yes.

I had everything right, with the exception of question 2, but the “correct” answer is faulty. One should see it as an syllogism which would make this:

statement 1: for all elements of X on earth Y is true

statement 2: X1 is on earth

conclusion Y1 is true

This is what we call formal logic. Although commonsense suggests otherwise, the statement of the mouse is true based on premises 1 and 2.

Nice little test, that does make a point. But I don’t agree with 3, 4, and 5 replies:

Number 3 reply should be 1/51 not 2/100. Out of 1,000, 1 will have it and 50 false positives, so that would be 51 positive tests.

On number 4 the rule says ‘If a card has a vowel on its letter side, it has an even number on its number side’ which doesn’t imply that every even number on a card translates into a vowel on the other side. So to test the rule, only card A should be turned.

On number 5, the answer should be no, because of lack of info for comparison. If you use, instead of ‘a certain German car’ ‘a Hummer car’, there is reasonable understanding that it will be tougher than a regular family car, not a reason to ban it…

Some of the comments regarding #4 are of the same nature as if the question was, “Here is a sequence. 0,1,2,3,4. What is the next number?” This question is basically unanswerable because the answer is, any number can be the next number. There is an assumption in most people’s mind that the question is “What is the simple rule behind the sequence and the problem is to determine the simple rule.” However the question in #4 is explicitly aimed at determining which of the given card(s) need to be turned over to verify or disprove the rule. Nothing more is asked for and no assumptions are required.

thanks, Number 5 Stumped me ” comprehensive study”

until I saw there was no right or wrong answer thanks nice test.. God Bless

#3 This isn’t a test of rationality — it is knowing how to calculate false positives. The false positive rate = false positive / (false positives + true negatives).

We know the false positive rate (.05) and the number of true negatives (999). Solving for the number of false positives gives us: 52.58 false positives (in a population of 1000). Adding the one true positive from the premise gives 53.58.

Thus the chances of actually having the disease if you get a positive result is: 1/53.58 or 1.86%.

The premise of the article is being rational. There is, however, nothing rational about arbitrary statistics that have specific meaning to statisticians. The rational part of me says this is a dumb question to test how rational people are.