Wason card problem

One of the nicer features of the James Randi Educational Foundation's Amazing Meeting earlier this year was the time set aside for mini-talks by those responding to a call for papers. One of those talks was given by Dr. Jeff Corey, who teaches experimental psychology at C. W. Post College. His talk was on "The Wason Card Problem" and its role in teaching critical thinking skills. Four cards are presented: A, B, 4, and 7. There is a letter on one side of each card and a number on the other side. Which card(s) must you turn over to determine whether the following statement is false? "If a card has a vowel on one side, then it has an even number on the other side."

(I suggest you spend a few minutes trying to solve the problem before continuing.)

(I hope you have been able to restrain yourself from jumping ahead and have worked out your solution to the problem. Before continuing, try to solve the following alternative version: Let the cards show "beer," "cola," "16 years," and "22 years." On one side of each card is the name of a drink; on the other side is the age of the drinker. What card(s) must be turned over to determine if the following statement is false? If a person is drinking beer, then the person is over 19-years-old.)

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I gave the Wason Card Problem to 100 students last semester and only seven got it right, which was about what was expected. There are various explanations for these results. One of the more common explanations is in terms of confirmation bias. This explanation is based on the fact that the majority of people think you must turn over cards A and 4, the vowel card and the even-number card. It is thought that those who would turn over these cards are thinking "I must turn over A to see if there is an even number on the other side and I must turn over the 4 to see if there is a vowel on the other side." Such thinking supposedly indicates that one is trying to confirm the statement If a card has a vowel on one side, then it has an even number on the other side. Presumably, one is thinking that if the statement cannot be confirmed, it must be false. This explanation then leads to the question: Why do most people try to confirm a statement, when the task is to determine if it is false? One explanation is that people tend to try to fit individual cases into patterns or rules. The problem with this explanation is that in this case we are instructed to find cases that don't fit the rule. Is there some sort of inherent resistance to such an activity? Are we so driven to fit individual cases to a rule that we can't even follow a simple instruction to find cases that don't fit the rule? Or, are we so driven that we tend to think that the best way to determine whether an instance does not fit a rule is to try to confirm it and if it can't be confirmed then, and only then, do we consider that the rule might be wrong?

Corey noted that when the problem is changed from abstract items, such as numbers and letters, and put in concrete terms, such as drinks and the age of the drinker, the success rate significantly increases (see the example described above). One would think that confirmation bias would lead most people to say they must turn over the beer card and the 22 card, but they don't. Most people see that the cola and 22 cards are irrelevant to solving the problem. If I remember correctly, Corey explained the difference in performance between the abstract and concrete versions of the problem in terms of evolutionary psychology: Humans are hardwired to solve practical, concrete problems, not abstract ones. To support his point, he says he simplified the abstract test to include only two cards (showing 1 and 2) with equally poor results.

I had discussed confirmation bias, but not conditional statements, with my classes before giving them the Wason problem. The majority seemed to understand confirmation bias; so, if the reason so many do so poorly on this problem is confirmation bias, then just knowing about confirmation bias is not much help in overcoming it as a hindrance to critical thinking. This is consistent with what I teach. Recognition of a hindrance is a necessary but not a sufficient condition for overcoming that hindrance. However, next semester I'm going to give my students the Wason test after I discuss determining the truth-value of conditional statements. The reason for doing so is that anyone who has studied the logic of conditional statements should know that a conditional statement is false if and only if the antecedent is true and the consequent is false. (The antecedent is the if statement; the consequent is the then statement.) So, the statement If a card has a vowel on one side, then it has an even number on the other side can only be false if the statement a card has a vowel on one side is true and the statement it has an even number on the other side is false. I must look at the card with the vowel showing to find out what is on the other side because it could be an odd number and thus would show me that the statement is false. I must also look at the card with the odd number to find out what is on the other side because it could be a vowel and thus would show me that the statement is false. I don't need to look at the card with the consonant because the statement I am testing has nothing to do with consonants. Nor do I need to look at the card with the even number showing because whether the other side has a vowel or a consonant will not help me determine whether the statement is false.

There is a possibility that the reason many think that the even-numbered card must be turned over is that they mistakenly think that the statement they are testing implies that if a card has an even number on one side then it cannot have a consonant on the other. In other words, it is possible that the high error rate is due to misunderstanding logical implication rather than confirmation bias. In the concrete version of the problem, perhaps it is much easier to see that the statement If a person is drinking beer, then the person is over 19-years-old does not imply that if a person is over 19 then they cannot be drinking cola. If this is the case, then an explanation in terms of the difference between contextual implication and logical implication might be better than one in terms of confirmation bias. Perhaps it is the context of drinking and age of the drinker that indicates to many people that a person can be over 19 and not drink beer without falsifying the statement being tested, i.e., that simply because if you're drinking beer you are over 19 doesn't imply that if you're over 19 you can't be drinking cola. That is, in the concrete case people may not have any better understanding of logical implication than they do in the abstract case and neither case may have anything to do with confirmation bias.

On the other hand, some might reason that if I turn over the even card and find a vowel, then I have confirmed the statement, which is in effect the same as showing that the statement is not false, but true. This would be classic confirmation bias. Finding an instance that confirms the rule does not prove the rule is true. But, finding one instance that disproves the rule shows that the rule is false.

The Wason Card Problem Revisited

I received several responses to my analysis of the Wason problem. Mathematician and author Jan Willem Nienhuys wrote from the Netherlands:

I don't think that the card problem as presented is compatible with the beer over 21 problem. What would happen if you said "vowels and odds are forbidden to go together on one card" and ask someone to check whether there are cards that are forbidden. That's the beer over 21 problem. Another problem with the example is that the beer problem has a known social setting. If you made some kind of funny restriction, like 'over 22 must drink coke', it's much harder, or you can make a restaurant setting, with a completely strange restriction like 'girls (or people with a polysyllabic name) must order broccoli', then it's much more difficult, for the problem solvers must then keep an odd fact in mind while analyzing several cases. The less unfamiliar facts one has to keep at same time ready in the mind, the easier it is. (And it is quite possible that not everybody knows what's an even number or what's a vowel, or that people with slightly deficient knowledge know at most one of these concepts; you'd be surprised how deficient people's knowledge is).
 

I replied to Jan that, unless I'm mistaken, both problems imply that two cards are forbidden together (vowel and odd number; beer and 19-years or under). I think I will try the problem on my classes with Jan's suggested instruction and see if the results vary significantly. (I'll send him the results and he, the mathematician, can tell me whether the difference, if any, is significant!) The social setting would be part of what I'm calling the context that might be why the beer problem is easier to solve for most people. It had not occurred to me that part of the problem might be in understanding the meaning of words like "vowel" and "even," but that is a consideration that should not be taken lightly (unfortunately) and maybe I should try the test with some set-up questions to make sure those taking it understand such terms.

Jan replied:

I will be very interested in what you find. You might try variations like: if there are two primes on one side, the other side must show their product. This means that if a card shows a single number that is the product of two primes, you don't have to turn it around. If it shows two numbers that aren't primes, you also don't have to turn it around. Obviously the difficulty is that lots of people don't know what are primes, and even if they do so theoretically, some know their tables of multiplication so poorly, that they are at loss what to do when the card shows 42 or 49 or 87 or 36 or 39. Or 10.

Yikes! Jan, I teach a general course in logic and critical thinking, not math! My students would lynch me if I posed such a problem to them.

I do think that one of the problems with solving this problem (and many others!) has to do with how one reads or misreads the instructions. (For those who don't recall the exact instructions, here they are again: Four cards are presented: A, D, 4, and 7. There is a letter on one side of each card and a number on the other side. Which card(s) must you turn over to determine whether the following statement is false? "If a card has a vowel on one side, then it has an even number on the other side."

One reader wrote:

My solution to the problem is to check all cards (or a random sample if there are a large number of them) - Sometimes it's best to see what rules apply. (Sometimes "if" means if and only if...)

This approach represents a common mistake in problem-solving: self-imposed rules. The instructions do not imply that there are more than four cards, nor does "if" mean "if and only if." (See James Adams' Conceptual Blockbusting for a good discussion on common hindrances to problem-solving.)

The reader continues:

A simpler explanation for people choosing A and 4: Given that people tend to satifice, it makes sense that many will just check the cards where they see a vowel or an even number. It's a quick solution made with the immediate data on hand, requiring no additional thought (about the implications of the statement or anything else). Classic satisficing behavior.

Whether this solution is satisficing or satificing, it's wrong.

Another reader, Jack Philley, wrote:

Thanks for a great newsletter. I am a safety engineer and incident investigator. I also teach a segment on critical thinking in my incident investigation course, and I have been using the Wason card challenge. I picked it up from Tom Gilovich's book How We Know What Isn't So. About 80 % of my students get it wrong and some of them become very angry and embarrassed and defend their logic to an unreasonable degree. I use it to illustrate our natural talent to try to prove a hypothesis and our weakness in thinking about how to disprove a suspected hypothesis. This comes in handy when trying to identify the actual accident scenario from a set of speculated possible cause scenarios.

For those who haven't read Gilovich (or have but don't remember what he said about the Wason problem), he thinks that people turn over card "2" even though it is uninformative and can only confirm the hypothesis because they are looking for evidence that would be consistent with the hypothesis rather than evidence which would be inconsistent with the hypothesis. He also finds this behavior most informative because it "makes it abundantly clear that the tendency to seek out information consistent with a hypothesis need not stem from any desire for the hypothesis to be true (33)." Who really cares what is true regarding vowels and numbers? Thus, the notion that we seek confirmatory evidence because we are trying to find support for things we want to be true is not supported by the typical results of the Wason test. People seek confirmatory evidence, according to Gilovich, because they think it is relevant.

As to the notion I put forth that it is because of the context that people do better when the problem is in terms of drinking beer or soda and age, Gilovich notes that only in contexts that invoke the notion of permission do we find improved performance (p. 34 note). This just shows, he thinks, that there are some situations where "people are not preoccupied with confirmations."

Last updated 12/09/10