## Tuesday, July 13, 2010

### Conjunction Fallacy

I have been reading Less Wrong

A blood clot on the lung often leads to shortness of breath, but rarely to weakness. So my sources inform me.

Imagine a man is suffering from a blood clot in the lung:

Rank in order of probability the following lists of symptoms:

A Weakness
B Calf pain
C Sharp pain while breathing
D Shortness of breath and weakness
E Loss of consciousness and fast heartbeat
F Coughing up blood

Actually do this, before reading on.

Did you rank D as more likely than A?

If not, congratulations. Medical students get this wrong, apparently. As do I.

If so, what are you thinking? Can you not see that weakness has to be more likely than weakness together with something else?

A person suffering from weakness and shortness of breath is suffering from weakness.

If out of 100 people with blood clots, 50 are suffering from weakness AND shortness of breath, then AT LEAST 50 must be suffering from weakness.

The thing is, I know about this effect, called the Conjunction Fallacy. And I know some probability and even a bit of statistics. And I still think that D outranks A. I couldn't be more surprised if I kept putting two oranges next to two oranges and getting three oranges.

Here's another one:

Suppose Venus Williams is playing tennis.

Rank in order the probability that:

A Venus loses the first set
B Venus wins the match
C Venus loses the first set and wins the match
D Venus wins the first set but loses the match
E Venus loses the match

Try this, without thinking too hard. No drawings of Venn diagrams are to be made.

I make this:

B>C>A>D>E

Venus seems likely to win, and it's also quite possible that she loses the first set but wins anyway.
It's quite unlikely that she loses the first set, but even more unlikely that she wins the first set but then loses the match (she'd have to lose two sets, and that's impossible. She's a goddess!). The idea of her losing seems very unlikely indeed.

How do you rank them? Same as me? If all our brains work like this then it's a wonder we ever manage to tie our shoelaces.

There's a lot of very good stuff along these lines at the Less Wrong wiki.

In the Venus example, it absolutely has to be true that A>C, B>C, and that E>D. I can see that, but I do not feel that.

I am never going to trust a hunch again. There is a name for my belief that hunches are right more often than they have any right to be.

What if we change the question?

Let's take a load of tennis matches, say the recent Wimbledon.

E. Take the loser from every ladies singles match result.
There'll be loads. One of them will be Venus, who didn't win the tournament.

D. Take the loser from every ladies singles match which was 2 sets to 1. There'll be fewer. I bet Venus is in there.

C. Take the winner from every ladies singles where the match was 2-1. There are the same number as for D. I bet Venus is in that list several times.

B. Take the winner from every ladies singles match. There'll be more, and I bet Venus is in that list more too, even proportionately.

A. Take every name of a woman who lost a first set. I have no intuition about how many times our hero is on that list relative to the others.

So E < D < C < B , and I have no idea where A should be.

This looks awfully like my intuitive answer to the first question. Even though it's the answer to a COMPLETELY DIFFERENT QUESTION.

Is there any way to avoid (or any reason to want to avoid) the conclusion that:

When you ask someone for the probability of A given B, they give you the probability of B given A.

Linda is a vegetarian who knits her own yoghurt and is active in the environmental movement.

What is the probability of her being a bank clerk?

What is the probability of her being a feminist bank clerk?

1. Re lung blood clot symptoms,I too think 'D' outranks 'A' and this because my instinct is to answer a different question i.e which of these conditions provide a stronger clue to the diagnosis - which has to be D. The number of conditions with 'Weakness' as a symptom are greater than the number of conditions with 'Weakness and Shortness of Breath' as symptoms.

2. Yes, I think that's the accepted explanation for the Conjunction Fallacy. It's know as the Representativeness Heuristic.

The probability of blood clots given shortness of breath and weakness is very different from the probability of shortness of breath and weakness given blood clots.

But we carelessly substitute one for the other even when we understand that that's not the question.

Apparently time pressure or high stakes mean that we get more likely to use the (wrong) shortcut. It's only when the problem doesn't feel important that we have any chance of reasoning rationally at all!

This is one of the reasons that probability theory/statistics is famously 'counter intuitive'. That is to say, our intuition is badly, consistently and demonstrably wrong.

3. I understand the difference. 'Short cutting' can lead us astray in the context of precisely phrased question, as it has here, but most dialogues are not set up this way and 'questions' are frequently imprecisely phrased. I like to see this instinct as an ability to see the better or intended question, and not always the wrong one. We need to be smart enough to know when to resist the short cut though – and a multiple choice probability question is a good example. Shame on us.

I have a more positive view of intuition. It has to be founded in a deep understanding of the issue at hand for it to be more than just a guess. When this is the case, there's complex form of logic at work. It’s not always expressible. How does the batsman know the nature of the next ball coming at him? I love the idea that he senses it in the bowler’s action but can’t explain how. Awesome.