# The Conjunction Fallacy

*You think your rational, right? Given all facts and enough time to decide, you can always come up with the correct solution?*

Well, really sorry to break it to you, but you are far from being a rational boolean agent! In fact, we humans are really bad at statistical decisions.

In this, and maybe some upcoming posts I will explain some cases where humans don’t act according to simple logic. These flaws can be avoided, once you are aware of them - so listen up! Obviously, the best approach is to test these fallacies on you, the reader:

So, assume the following description of *Heiner *is true. Given these facts, order the 6 options below **by their probability** meaning the most probable option first and then ending with the least probable option. No complaints about the description, please, this experiment was done in 1974.

*Heiner is 34 years old. He is intelligent but unimaginative, compulsive, and generally lifeless. In school, he was strong in mathematics but weak in social studies and humanities.*

A: Heiner is an accountant.

B: Heiner is a physician who plays poker for a hobby.

C: Heiner plays jazz for a hobby.

D: Heiner is an architect.

E: Heiner is an accountant who plays jazz for a hobby.

F: Heiner climbs mountains for a hobby.

Take a moment to rank these six propositions by probability. Write them down so you can`t cheat.

In a very similar experiment conducted by Tversky and Kahneman in 1982, 92% of 94 undergraduates at a well-known American university gave an ordering with **A** > **E** > **C**, did you do too? The ranking **E** > **C** was also displayed by 83% of 32 grad students in the decision science program of Stanford Business School, all of whom had taken advanced courses in probability and statistics.

There is a certain **logical problem** with saying that Heiner is more likely to be “an accountant who plays jazz” than he is to only “play jazz”. The conjunction rule of probability theory states that, for all X and Y, P(X&Y) <= P(Y). That is, the probability that X and Y are simultaneously true, is always less than or equal to the probability that Y is true. Violating this rule is called a **conjunction fallacy.**

Imagine a group of 100,000 people, all of whom fit Heiner’s description (except for the name, perhaps). If you take the subset of all these persons who play jazz, and the subset of all these persons who play jazz *and* are accountants, the second subset will *always* be smaller because it is strictly contained within the first subset.

Why would highly educated students with knowledge of statistics still fail this test? Why did you fail the test even though you knew it was a test? One explanation would be a misunderstanding of the statements, a problem with wording and framing. Maybe you understood “A: Heiner is an accountant” as “Heiner is an accountant but he doesn’t play Jazz”.

It is then possible that “E: accountant & plays jazz” > “A: accountant & doesn’t play Jazz”. Another problem is that many people will maybe mix probability and plausibility — meanings “what is plausible”, and “whether there is evidence”. But we’ll talk about this later, but first some more tests.

Tversky and Kahneman (1983), played undergraduates at UBC and Stanford for real money:

*Consider a regular six-sided die with **four green** faces and **two red** faces. The die will be rolled 20 times and the sequences of greens (G) and reds (R) will be recorded. You are asked to select one sequence, from a set of three, and you will win $25 if the sequence you chose appears on successive rolls of the die.*

**1**. RGRRR**2**. GRGRRR**3**. GRRRRR

65% of the subjects chose sequence **2**, which is most representative of the die since the die is mostly green and sequence **2** contains the greatest proportion of green rolls. However, sequence **1** *dominates* sequence **2**, because sequence **1** is strictly included in **2**.

This clears up possible misunderstandings of “probability” as stated above since the goal was simply to get the $25.

Another experiment from Tversky and Kahneman (1983) was conducted at the Second International Congress on Forecasting in July of 1982. The experimental subjects were 115 professional analysts, employed by industry, universities, or research institutes. Two *different* experimental groups were respectively asked to rate the probability of two different statements, each group seeing only one statement:

*“A complete suspension of diplomatic relations between the USA and the Soviet Union, sometime in 1983.”*

*“A Russian invasion of Poland, and a complete suspension of diplomatic relations between the USA and the Soviet Union, sometime in 1983.”*

Estimates of probability were low for both statements but significantly lower for the first group (1%) than the second (4%). Since each experimental group only saw one statement, there is no possibility that the first group interpreted (1) to mean “suspension but no invasion”*. *With these last two examples, we have found evidence against our “explanations” of example one.

**So, adding more detail or extra assumptions can make an event seem more plausible, even though the event necessarily becomes less probable.**

There are many other fallacies and psychological effects that show how limited and non-bayesian our brain actually is. However, this one really hit me hard!

Take a bit of time the next days to figure out where this effect is important. When a politician tells you how he wants to achieve something it seems more reachable, right? In politics and economics predicting the future is important but basically, no one is good at it. By adding details and explanations everything seems more probable. Especially the last test example is impressive. It almost seems like our brain is happy not to think about why something will happen and happily accepts the explanation of a polish invasion, even though this makes the whole statement less probable!

The reason why this fallacy is growing in relevance is this new technology, the internet (Neuland).

Thanks to the vast amount of information (or misinformation) on the internet, we are all able to build stories, ideas, and “facts” on the fly. Nowadays, we can all easily be misguided by confirmation bias, our natural tendency to search for information that confirms our beliefs and to ignore that which threatens our beliefs.

The problem is that the abundance of blocks makes it very easy to string together stories supporting A, B, and C. The internet makes readily accessible vast numbers of marginally relevant or manifestly false details about almost everything and everybody. Because of this, confirmation bias and the conjunction fallacy are very easy traps to fall into. Untrue stories are believable not only because of our partisanship and our confirmation bias but also because of the proliferation of information on the internet. Anyone can now pick and choose from the internet’s vast trove of “facts” to colorfully embellish a simple story and make it superficially plausible.

*This in itself is a whole other topic I should maybe not squeeze into this post, but anyway, can’t erase it now…*

There are many, many other situations in life the conjunction fallacy applies to. So the next time you are tempted to believe something or someone, trust mathematics not yourself!

**And as always, stay curious!**