Linda, 31, is a philosophy major who went to Berkeley. She is deeply concerned with social justice and discrimination and participated in antinuclear demonstrations. Which alternative is more likely: a) She is a bank teller; or b) She is a bank teller and an activist in the feminist movement?Discussion below the fold.The authors note that 85% of those asked this question made the "boneheaded error in basic logic" of choosing the second option. Why is it boneheaded? Because the second option is a subset of the first option, and therefore cannot be more likely.
Why not? Let's say I have a bag with a hundred marbles in it, 35 red and 65 black. Of the red marbles, some have a green spot on them and others do not. Is it more likely that you will pull out a red marble, or a red marble with a green spot? Logically, even if 34 of the 35 red marbles have green spots, it will be less likely because the green-spot-reds are a subset of the reds. Even if all of the reds have green spots, it cannot be more likely that you will pull out a green-spot-red than a red.
The process that is fooling the test-takers is that they are considering the question as two separate cases: Is it more likely that she is a bank teller, or a feminist activist? There's no special reason to believe a philosophy major (let alone one concerned with 'social justice and discrimination') will become a bank teller; so it seems far more likely that she is an activist. Thus, the option that includes the higher-probability choice seems more probable overall -- even though, in fact, it can't be.
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A reminder that even relatively simple science (like relatively simple economics) can pretty quickly get you into territory where intuition will lead you astray. And sometimes mathematics is the guide that will take you further.
It's a danger even outside of mathematics, too. The other day we saw a video here in which a guy asks a fellow, "How do you spell shop?"
"Shop?"
"Yes, shop."
"S-h-o-p."
"What do you do when you come to a green light?"
"Stop."
"At a green light?"
Intuition can be very helpful in some circumstances, but there are some holes in it. Someone who knows how can use them to lead people -- let's say 85% of people -- astray.
On the other hand, there are those whose intuitions run opposite. I would have gotten this question right even without the math, because what my intuition said was that it sounded like a trap.
Well, wait- If I'd been asked the question with it set up as 'assess the probability', I'd have given the correct answer. In casual reading, I would likely make the incorrect answer, but I wouldn't have thought of it as binding or important. The thing is, we're discussing probability, and probability isn't about reality, it's about the math of things. As you pointed out, "Even if all of the reds have green spots, it cannot be more likely that you will pull out a green-spot-red than a red." Now, in the real world, that's obviously not true- but if we're talking about prediction and probability in the abstract, logic says you're correct. I think that's a critical distinction. When you add in the factors that drive people to make decisions about their lives, like being an activist or not, probability is even less precise a predictor, because those choices weren't made strictly logically in the first place.
Now, I'm not surprised that you'd get it right- you are well versed in the practice of logic.
The question is about probability ("what is more likely?") but it may be the test isn't quite fair. They are right to say that (b) is a subset of (a), but the question is made to sound as if the question was not about a subset but rather an exclusive disjunction. In other words, it sounds like they're telling you that she's either a bank teller who is not an activist, or a bank teller who is also an activist.
In that case, the instinct could be correct -- here the analogy would have yellow dots on some of the red marbles, and green dots on others. If there are 10 red marbles with yellow dots and 25 with green dots, then yes, (b) is more likely than (a).
The problem is that they aren't asking you about an alternative subset (e.g., "(a) a bank teller who is not a feminist activist, or (b) a bank teller who is a feminist activist"). They're asking you about the set proper versus a subset.
Sure, it's just that I don't know if when the question was asked of the college students, they were aware of the framing of the question as one of probability (as in the mathematical, as opposed to synonym for likelyhood sense). If they were not clearly advised that was the case, I think it would be an invalid study.
I suppose also, while I agree with you that it's important to understand probability and apply that knowledge to the information we encounter, we also need to understand the limits of probability- for instance, the case of the red and black marbles you used in the original post that would be logically correct in the mathematical sense of probability, but completely wrong in a real world sense.
The reason it tripped up the students is that they subconsciously inserted a NOT into the comparison. If this was normal conversation, that would be justified. Then, it is A NOT B vs. A AND B.
It is more about linguistic assumptions than about logic. It is akin how saying A or B in general conversation carries NOT A AND B along with it.
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