"Erroneous" Probabilistic Reasoning

I’ve been reading Leonard Mlodinow’s “The Drunkard’s Walk: How Randomness Rules Our Lives”, and he describes a set of experiments which I had heard of before but never gave too much thought to. The experiments deal with people making probability assessments about a series of statements. The experiments were done by Daniel Kahneman and Amos Tversky[cite here]. It starts with a description:

Imagine a woman named Linda, thirty-two years old, single, out-spoken, and very bright. In college she majored in philosophy. While a student she was deeply concerned with discrimination and social justice and participated in antinuclear demonstrations.
They then ask for a ranking of most (1) to least (8) probable for a number of statements. The interesting three statements are:
Linda is active in the feminist movement: 2.1
Linda is a bank teller and is active in the feminist movement: 4.1
Linda is a bank teller: 6.2
This is then used to say that people do not figure probabilities correctly because “the probability that two events will both occur can never be greater than the probability that each will occur individually” (italics in original).
The book reports that “even highly trained doctors make this error”, with the following example.

They presented a group of internists with a serious medical problem: a pulmonary embolism (a blood clot in the lung). If you have that ailment, you might display one or more of a set of symptoms. Some of those symptoms, such as partial paralysis, are uncommon; others, such as shortness of breath, are probable. Which is more likely: that the victim of an embolism will experience partial paralysis or that the victim will experience both partial paralysis and shortness of breath? Kahneman and Tversky found that 91 percent of the doctors believed a clot was less likely to cause just a rare symptom than it was to cause a combination of the rare symptom and a common one. (In the doctor’s defense, patients don’t walk into their offices and say things like “I have a blood clot in my lungs. Guess my symptoms.”

Now, I haven’t read past this point, or the original study, so take what I say here with a grain of salt. I wanted to put down my thoughts on these observations before going on to read the study’s conclusions. Perhaps what I say now will be inconsistent with other aspects of the studies, or further data.
I do not think that one should conclude poor reasoning in these examples.
I believe there are two things going on here. One is a property of the English language, and the other is a property of human reasoning. In English, if I were to say “Do you want steak for dinner, or steak and potatoes?” one would immediately parse this as choice between
1. steak with no potatoes
2. steak with potatoes
Although strict logic would have it otherwise, it is common in English to have the implied negative when given a choice like this. If we interpret the doctor’s choice, we have:
1. clot with paralysis and shortness of breath
2. clot with paralysis and no shortness of breath
the second one is much less likely, because it would be odd to have a clot and not have a very common symptom associated with it. It is less clear in Linda’s case, but I think the same reasoning applies there. What is interesting is that the error is not seen in ranking statements which have nothing to do with the given knowledge about Linda, such as:
Linda owns an IHOP franchise
Linda had a sex-change and is now Larry
Linda had a sex-change and is now Larry and owns an IHOP franchise
There might be something to being completely unrelated that changes the interpretation of the English sentence, and makes it a bit more formal, closer to the mathematical reasoning. I am not sure what types of statements would do this, but it is a bit challenging to disentangle subtle language interpretations I think.
When reading these experiments, I recalled a description from E.T. Jaynes about people receiving the same new information, but updating their knowledge in a diverging way, due to differences in their prior information. I think something like that could be going on here. What I mean is, when doctors are asked: “Which is more likely: that the victim of an embolism will experience partial paralysis or that the victim will experience both partial paralysis and shortness of breath?” it is interpreted as:
1. someone is claiming that the patient has an embolism
2. the patient is claiming, or someone has measured, that she has partial paralysis
3. the patient is claiming, or someone has measured, that she has shortness of breath
I don’t believe the doctors are separating the analysis of the claim of the clot, which is given information, from the other claims. As Mlodinow admits, the situation where one knows the diagnosis is practically never encountered, so the doctors are really assessing the truthfulness of the existence of the clot. Because of this, the implied negative in (2) above (i.e. paralysis with no shortness of breath) is even stronger.
Another way of looking at it is to include the knowledge of the method of reporting. Someone who is reporting information about an ailment will report all of the information accessible to them. By reporting only the paralysis, there are two possibilities concerning the person measuring the symptoms of the patient:
1. they had the means to measure shortness breath in the patient, but there was none
2. they did not have the means to measure shortness of breath
In the first case, the doctor’s probability assessment is absolutely correct: both symptoms together are more likely than just one. In the second case, the doctors are also correct: one of the sets of diagnostic results (i.e. just paralysis) is less dependable than the other set (i.e. both symptoms), thus the second one is more likely to indicate a clot or is consistent with the known clot.
It isn’t that the doctors are reasoning incorrectly. They are including more information, and doing a more sophisticated inference than the strict, formal, minimalistic interpretation of the statements would lead one to do.
This analysis works well for other examples stated in the book, like “Is it more probable that the president will increase federal aid to education or that he or she will increase federal aid to education with function freed by cutting other aid to states?”.
Now I have to continue reading the book, and track down the study, to see if any of these thoughts pan out.

I am a professor of Science and Technology at Bryant University in Smithfield, RI, and a research professor in the Institute for Brain and Neural Systems, Brown University. My research is in computational neuroscience and statistics. I teach physics, meteorology, astonomy, theoretical neuroscience, systems dynamics, artificial intelligence and robotics. My book, "Theory of Cortical Plasticity" (World Scientific, 2004), details a theory of learning and memory in the cortex, and presents the consequences and predictions of the theory. I am an avid python enthusiast, and a Bayesian (a la E. T. Jaynes), and love music.
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2 Responses to "Erroneous" Probabilistic Reasoning

1. How refreshing to find your comment. I'm reading the same book and got really confused. Let me explain. If the probability of a partial paralysis is p(P.P.) and the probaility of shortness of breath is p(S.B.), then p(P.P.) = p(P.P. and S.B) + p(P.P. and NOT S.B.). Then he presents the problem to the doctor in one way : “Which is more likely: that the victim of an embolism will experience partial paralysis or that the victim will experience both partial paralysis and shortness of breath?”
So, obviously p(P.P.) >= p(P.P. and S.B). But then he says the doctors are wrong to think “a clot was less likely to cause just a rare symptom than it was to cause a combination of the rare symptom and a common one”. So, now he is comparing p(P.P. and S.B) and p(P.P. and NOT S.B). Well, arithmetically speaking, we don't know which one is bigger ! Actually, if I was to guess, assuming that P.P. is rare and S.B. is common, the equation is probably something like :
p(P.P.) = p(P.P. and S.B) + p(P.P. and NOT S.B.)
0,10 = 0,09 + 0,01
I was confused by the choice of words, but you took the line of thought further. Thanks !

2. bblais says:

“How refreshing to find your comment. ” – thanks!

“obviously p(P.P.) >= p(P.P. and S.B)” – correct.

“But then he says the doctors are wrong to think “a clot was less likely to cause just a rare symptom than it was to cause a combination of the rare symptom and a common one”. So, now he is comparing p(P.P. and S.B) and p(P.P. and NOT S.B). “

I think the author is claiming that he is comparing p(P.P) with p(P.P. and S.B), but as I wrote above, the way that English works there is an implied negative (like the steak and potatoes). The author thinks that the doctors are comparing p(P.P) with p(P.P. and S.B) but what is more likely is that the doctors, as you say, are comparing p(P.P. and S.B) and p(P.P. and NOT S.B). Thus the author is incorrect when he attributes poor reasoning skills to the doctors. We could just as easily attribute poor English skills to the author.

“Actually, if I was to guess, assuming that P.P. is rare and S.B. is common, the equation is probably something like :
p(P.P.) = p(P.P. and S.B) + p(P.P. and NOT S.B.)
0,10 = 0,09 + 0,01 “

In the book, the scenario is stated that we are given that the patient has a blood clot. So we are really looking at:

p(P.P and S.B | clot)

and

p(P.P and NOT S.B | clot)

with the latter probability being much smaller than the former (given that you have a clot, then you almost certainly will have shortness of breath).