Accidental Blogger

The classic demonstrator of the conjunction fallacy is the following:

1. Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations. Which is more probable?
1A. Linda is a bank teller.
1B. Linda is a bank teller and is active in the feminist movement.

The general form is this: of an entity X, qualities x_i are asserted. We are to decide whether X is more likely to possess quality y or quality y in addition to to some quality z. The fallacy consists in the fact that people, when queried, pick 1B. as more probable, when, as a proper subset of 1A., it couldn’t possibly be. (All feminist bank tellers are bank tellers but some bank tellers aren’t feminist) Now, we commonly understand this as emerging from use of the heuristic of representativeness. To use the jargon of the election cycle, latte-sippers like Linda sound more like feminists than like bank-tellers. So much more so, in fact, that people are willing to contemplate and assert mathematical impossibilities like p(y & z | x_i) > p(y | x_i) simply because p(z | x_i) > p(y | x_i).(*) Here, the x_i‘s are the facts stated about Linda, y is the state of her being a bank teller and z is the state of her being a feminist.

Okay. But now consider a related but different situation:

2. Someone murders a nice Jewish couple in a high-rise apartment complex. Linda is a suspect. In which of the following situations is it more appropriate to convict her? She’s seen to leave the building:
2A. five minutes later
2B. five minutes later, covered in blood, with a gun in her hand, and with a swastika inked on her exposed right forearm.

Now here too 2B is a proper subset of 2A, yet now the the correct response is obviously 2B, not 2A. There’s no logical puzzle here; it’s just that now we’re reasoning backwards from the observations 2A, 2B to settle the truth of 2. That is, now we’re comparing p(x|y & z) to p(x|y) and it’s quite kosher for the first of those two to be larger (or smaller) than the second.

Now for the musings: in addition to failures of the representativeness heuristic, does some part of the conjunction fallacy arise from matters of this kind? After all,
– we pretty generically confuse the p(A|B) with p(B|A) – assuming we intuit that these things are different at all.
– we often confuse the thing assumed with the thing to be proved, as anyone who’s ever tried an A-only-if-B style math proof knows too well.

Maybe instead of following imperfect heuristics through to the implicit conclusion that intersections can be more probable than what they’re intersections of, some people are just trying to “convict” Linda of her stated biography (1) in two alternate worlds 1A and 1B, and deciding that 1B gives a more workable case for the prosecution. If so, their reasoning wouldn’t instantiate idiotic math, just math that – reliably and well – answers a different question. Maybe the very fact that respondents are being asked about these things using the language of probability pushes some of them to start thinking like good prosecutors, getting them to condition priors on posteriors

How might we test for such? It’s hard, and I wouldn’t know, but here’s the first thing I would try: ask people question 1. as before, except ask them to assign numerical subjective probabilities to options 1A and 1B. Then, ask them to assign probabilities to:
1C. Linda is a feminist.

1D. Linda is anti-nuclear.

I’d expect that at least some of the people who committed the conjunction fallacy would be tricked into assigning lower probability to 1D. than to 1C, something which no-one who saw 1D as given fact should do.

Finally, what if some people, instead of reversing premises and conclusions, aren’t separating them clearly at all? What if what’s being done instead is to evaluate an overall web of interrelated fact-claims for how well it hangs together? This would be representativeness heuristic taken to a certain logical endpoint, where all the different qualities mentioned simply may or may not apply to someone labeled ‘Linda’, and one essentially evaluates each belief for plausibility in the background of the rest. To test for that, one might ask something like this:

3. Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy at Vassar College. She intends to home-school her children some day. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations. Which is more probable?
3A. Linda is a bank teller.
3B. Linda is a bank teller and is active in the feminist movement.

If some people really are doing this last horrible(**) thing, I’d expect such to assign measurably lower probabilities to:
3C. Linda approves of home-schooling
than to
3D. Linda is anti-nuclear.
(She’s been plonked into Vassar to “compensate” for the intuition that feminists are less likely to homeschool)

Conclude ramble.

(*)I’m setting entirely to the side the question of whether knowing a person is like Linda in fact raises the odds of her being a feminist more than it does those of her being a bank-teller.

(*) Well, horrible in this kind of setting at any rate, where the known is utterly known, beyond all possible doubt and whatnot. In more realistic situations, we frequently must revise assumptions in light of conclusions, and reason our way out of particular first principles and into others. Maybe Linda wasn’t that big on social justice after all. Many people simply mayn’t have acquired the peculiar academic discipline of really, truly, completely, suspending disbelief when presented with a set of consistent assumptions and therefore end up thinking about them instead of just with them. It’s not like we particularly excel at abstract thought anyway…