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:
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:
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?