# The conjunction fallacy

Rob Brooks asks you to take a pop quiz:

When Jack was young, he began inflicting harm on animals. It started with just pulling the wings off flies, but eventually progressed to torturing squirrels and stray cats in his neighbourhood.

As an adult, Jack found that he did not get much thrill from harming animals, so he began hurting people instead. He has killed 5 homeless people that he abducted from poor neighbourhoods in his home city. Their dismembered bodies are currently buried in his basement.

Now, knowing what I have just told you about Jack, is it more probable that Jack is: A) A teacher. Or B) A teacher who does not believe in God?

The answer is of course A. It has nothing to do with the merits of being a teacher or an atheist or any of the particulars of the case. It is purely a matter of probability. If you have two sets of events consisting of one set A and another set B that is a subset of A, then B is always less likely than A because if you belong to the set B then you automatically belong to the set A but not vice versa.

But some people may reason in a different way because of what is known as the conjunction fallacy. They start out by viewing the act in moral terms and if they think that atheists are less moral or are amoral, then they may be drawn to choose option B, seeing the information about teachers being common to both cases and thus irrelevant. In fact, if you add an option C that says “A teacher who does not believe in God and also poisons stray animals”, people may think that is even more likely since it adds weight to the moral argument.

Daniel Kahneman in his excellent book Thinking Fast and Slow that I discussed back in 2012 uses many examples like this to illustrate the difference between System I thinking and System 2 thinking. “System 1 operates automatically and quickly, with little or no effort and no sense of voluntary control” while “System 2 allocates attention to the effortful mental activities that demand it, including complex computations. The operations of System 2 are often associated with the subjective experience of agency, choice, and concentration.” (p. 20)

The trouble is that we System 1 always kicks into action first because it is easy and automatic but it can lead us astray. It takes practice to rein it in and give time for System 2 to kick in before making a judgment.

1. Rob Grigjanis says

The name Kahneman didn’t ring a bell, but the types of thinking did. I saw a talk on TVO (TV Ontario) a couple of years ago about this. Lo and behold, it was Kahneman! Interesting talk.

2. Trebuchet says

I believe I saw this quiz on another FTBlog a year or so ago. Worth repeating anyhow.

3. Seth says

“If you have two sets of events consisting of one set A and another set B that is a subset of A, then B is always less likely than A because if you belong to the set B then you automatically belong to the set A but not vice versa.”

This isn’t exactly correct, though the spirit of it is. If, for example, all teachers were atheists, then the probability that someone is a teacher *and* an atheist would be equal to the probability that one was a teacher. In this instance, your ‘A’ is the statement ‘subject is a teacher’ and your ‘B’ is the statement ‘subject is a teacher *and* an atheist’.

A better way of constructing this argument is to have A = ‘subject is a teacher’, B = ‘subject is an atheist’, and C be the conjunction of A and B. Then A is obviously incomparable to B (I am an atheist, for example, and only in a very limited way might I be considered a teacher, whereas there are several credentialed teachers who are fundamentalist religious people). But C is a (proper) subset of both A and B. Then P(C) is strictly smaller than either P(A) or P(B).

However, if A and B are comparable (i.e., one is a subset of the other, say B is contained inside A), then their intersection is simply equal to the smaller set (by our supposition, C = B). In this instance, the probability of the conjunction is equal to the probability of one of the antecedent conditions.

So, yes, in most cases (almost all, in the technical and layman sense of that phrase), the conjunction probability is strictly less than either of the antecedent probabilities. But to be able to make that determination, you must be able to justify why the antecedent conditions each properly contain the conjunction (i.e., that C is a *proper* subset of A and B). It is obvious in the case under consideration, but it is not always so obvious.

4. fkloogok says

I don’t think this has anything to do with probability. We are mind-readers, and if some information is missing in communication we add it ourselves. In this example, we assume the author of the question for some reason forgot to add ‘a teacher who believes in god’, or maybe he thought it would be obvious what he was asking once we read choice b and didn’t care to specify it in each choice, since it kinda makes sense what he’s asking once we’ve read both choices.

And then after that little addage of our own, we make the choice most favorable to ourself.

The problem here is language, it doesn’t translate well to logical thinking, unless you severely restrict it and make the rules clear as glass. ‘Probable’ here in this sentence, doesn’t read as ‘mathematically probable’, but as ‘probable according to your opinion’.

5. lorn says

Actually I was thinking that ‘A’ was the more likely.

Mass murder and torture to “feel” is an activity that needs an excuse. The most common excuses seem to be: ‘A supernatural power told me to do it so it is okay’ and ‘God made me this way for a reason so it is okay’. Of course, once God enters the discussion there is no more real debate or seeking of answers. Willing and cooperative or shocked and objecting the person contextualizes themselves as the helpless puppet of a supernatural power.

IMO an atheist would be more likely to keep asking questions and to seek to modify the behavior. They may not actually be able to change the behavior but they are more likely to keep asking questions and struggling with the issue simply because, for them, there is no supernatural forces mandating their creation or actions.

I would like to think that a atheist and skeptic would try to place his tendencies in terms of a wider context and would observe that what was feeling right and good to him was wrong and evil to the victims and surrounding society. Eventually he realizes that it is self-flattery and indulgence which is motivating the torture and, at the very least, he killing could be placed in a more useful context.

Essentially you end up with Dexter. A cable TV series well worth watching. The story is about about a psychologically damaged kid adopted by a cop who teaches him a set of rules that limits who he can kill and how and rules that keep him from getting caught as he works for the Miami police department as a forensic specialist. A police procedural with a twist.

6. says

This area fascinates me greatly. Understanding the machinery that generates cognitive bias and resulting logical fallacies is very valuable in all sorts of interactions and encounters.

Cognitive debiasing is another interesting area.
http://www.improvediagnosis.org/blogpost/950784/170746/Cognitive-Debiasing

It’s based on removing bias in medicine, but the applications can be extended to all sorts of other areas. Now if only we had a better way of sorting all the interactions that are involved in generating and producing the dynamics political behavior and we can make shredding bad arguments and the behaviors of people with no arguments and all emotion more formula based.

7. StevoR : Free West Papua, free Tibet, let the Chagossians return! says

Now, knowing what I have just told you about Jack, is it more probable that Jack is: A) A teacher. Or B) A teacher who does not believe in God?

The answer is actually C) Neither a teacher nor an atheist teacher.

I find it hard to imagine a serial killer or someone like Jack is described as being wold either want or qualify to be or end up being employed as a teacher.

Regardless there is insufficient evidence to suggest any sort of teaching career is likely here nor does the information have any bearing on this characters religious beliefs or lack thereof.

But…if two people were equally inclined to kill, and one truly believed in the fiction of a divine god, wouldn’t the dynamics of their continued behavior be different?

The believer has a stark choice, repent and stop, or burn in hell. The atheist on the other hand could rationally construct reasons why killing was OK (we do this every day, with war and the death penalty).

It seems to me that the model of the human is what is different…in one case, a primal moron, in the other case a calculating thinker….and the outcome would actually be situational.

9. Marshall says

@SteveoR the question asked which was more probably, not whether that probability was high. C is not correct, since A has a higher probability than B, regardless of how low these probabilities are.

10. StevoR : Free West Papua, free Tibet, let the Chagossians return! says

@ ^ Marshall : Guess I’m just pointing out that there are other more likely options and the scenario as listed doesn’t give much if any reason to think either A or B are right.

11. Holms says

The believer has a stark choice, repent and stop, or burn in hell. The atheist on the other hand could rationally construct reasons why killing was OK (we do this every day, with war and the death penalty).

You appear to be unaware of the fact that the death penalty is more heavily supported by the religious than the secular… and the same can be said for violence in general, with the religious jail inmates disproportionately outnumbering the secular inmates.

Anyway, the logic works like this. Even if we assume for a moment that atheists are more likely to match this constructed scenario than he religious, the question is not asking us to compare [teachers (atheists)] to [teachers (relgious)]; rather, the question is asking us to compare [teachers (atheists)] to [teachers (all subsets combined)]. The general category is always going to be more populated and thus the more likely candidate, because it includes [teachers (atheist)] AND all other populations.

12. says

But…if two people were equally inclined to kill, and one truly believed in the fiction of a divine god, wouldn’t the dynamics of their continued behavior be different?

Sure, based on their own practice at being rational and logical in decision making and there is no way to determine that. I would expect a somewhat higher chance of the atheist being rational (but not much) and logical because that can be involved in the decision to become an atheist, but that is not guarantee. There are plenty of atheists out there that are less rational and logical than many christians.

To make matters more complicated each individual is likely differently rational and logical on different issues depending on their emotional investment in particular issues, and practice being careful about issues that they know they are invested in.

The believer has a stark choice, repent and stop, or burn in hell. The atheist on the other hand could rationally construct reasons why killing was OK (we do this every day, with war and the death penalty).

Or the believer thinks they are ordered to kill. Or the atheist is an atheist because of how they were treated by religion and they rationalize killing for some other emotionally based reason (threatening to one’s society, politics, other terrible reasons that history has seen before). The reasoning born of our cognitive biases is the default condition. You have to teach rationality and logic, they are tools every bit as much as the scientific method.

It seems to me that the model of the human is what is different…in one case, a primal moron, in the other case a calculating thinker….and the outcome would actually be situational.

It’s not a “primal moron” and it’s just as useful to a person who is a calculated thinker. A better way of thinking about it is stored rapid “in-the-moment” responses that don’t need concentration (system one), versus careful analysis that one can use to modify the rapid responses (system two).

The default programming involves instincts that are aimed at “winning” instead of being correct and consistent with reality, which would have been much more helpful in life-and-death decisions through out human history and evolution, until more recently. This default programming causes people to learn to define things by relationship with personal and family social groups (or associated symbols) rather than what the things are on their own, and try to win on emotional political points instead of an assessment of facts of reality. We all know things either because we have investigated them first hand, or we have impressions about them because of people we know and trust. We can know everything so cognitive biases take over when we need to make decisions in the absence of personal knowledge.

For example the conservative obsession with making Obama into a Hitlarian monster, or desperately trying to distract things politically with Benghazi (won’t someone think of the dead soldiers?), or the birthers (he’s not one of us!), or responses to fixing health care (socialism!), or fixing the economy (socialism/communism!). The left has it’s examples too and they involve the balance between believing because your social group “know” (think about the usefulness of echo chambers), or knowing because you investigated it, and complimenting both with reasoning and logic skills.

13. Paul Jarc says

fkloogok #4, that objection works for this particular demonstration of the conjunction fallacy, but other demonstrations have been constructed to answer that objection. This one is an in-subject experiment, where one person is presented with both options, and we compare their responses to the two. Some other experiments are between-subjects, where each person sees only one option and gives an estimated probability. So there’s no opportunity to reinterpret one option in the context of the other. Then, for a large sample of subjects, we can compare the average probability estimates of the two options, and still see the same effect: people are asked to evaluate probability, but they implicitly substitute an evaluation of resemblance. Adding detail can increase resemblance, but it can only lower probability.

14. smrnda says

I always thought these were just a case where people mistakenly use “the probability of B GIVEN A” when the actual probability being evaluated is “the probability of B AND A”

15. Jockaira says

Now, knowing what I have just told you about Jack, is it more probable that Jack is: A) A teacher. Or B) A teacher who does not believe in God?

I think it more likely that Jack is getting a powerful assist from God Hisself, or may be the Devil incarnate.

Have you ever tried to catch a squirrel? Where do you think we get the expression “squirrelly” from?

By the way, even if you manage somehow to captue one, don’t even think of torturing it…you may end up minus a finger or two.