John Pinkerton writes (on Facebook),
I always interpreted the bank teller probability as “If I were to tell you that Linda is a bank teller and is active in the feminist movement, how likely would you think I was correct?” For this meaning, if you describe someone in greater detail, it’s more likely you know them well and thus describe them accurately.
This is a plausible explanation. However, I have seen students wrestle with such problems when they are presented as a question of whether the sequence of coin flips HHT has higher probability than the sequence HHTT. Pinkerton’s comment is on Steven Poole’s article. About the bank teller example, Poole writes
Tellingly, the psychologists Ralph Hertwig and Gerd Gigerenzer reported in 1999 that when you give people the same puzzle and ask them to guess about relative frequencies instead of what is more ‘probable’, they give the mathematically correct answer much more often. One might add that, if we are talking plausibility, the notion that Linda is a bank teller and an active feminist fits the whole story better. Arguably, therefore, it is a perfectly rational inference: all the available information is now consistent.
The entire article is recommended.
I thought his comment about “climate science” was worth noting:
“Of course, when one combines what Kahan stresses are ‘individually rational’ decisions into a group belief, one might judge that the group as a whole is being irrational in rejecting robust scientific evidence. ”
Given your blog post about science a few days ago, it is odd that he talks about “robust scientific evidence” for global warming.
It seems the inference of his data depends on your assumptions. If you assume people – on average – answer the question that was asked then you must infer that we are bad at basic probability/logic/set theory. If you assume that we get basic probability/logic/set theory then the inference is that people are answering different questions than the ones asked. Hertwig and Gigerenzer support the latter.
I would think there is a joint distribution, with some being good at one or the other, and some being good at both. I would think that by and large people struggle with problems like Monty Hall because they are answering the wrong question (Monty selects one box at random) for instance. I would guess what your students struggle because they are answering a question about frequency, not sequence. Some people who are bad at both logic and comprehension might even get the right answer for the wrong reason. It seems tough to generalize about the marginal distributions of each factor.
Part of the problem is that the language of probability and statistics overlaps with colloquial language in a contradictory way.
For example, if you have a wildly skewed distribution like lottery payouts, the “Expected Value” as calculated by formulas in a textbook has nothing at all to do with what a reasonable person would *expect to happen*. Economists, despite having the same problem with their own terminology (see: “Demand”), seem to get muddled by this. They say that under probabilistic decision-making conditions that if a person chooses anything other than what maximizes Expected Value that they’re being irrational. This is plainly nonsense.
Willy Wonka has placed five golden tickets in candy bars to be sold in June, July, and August. Each month will see 10 million candy bars sold. If Charlie’s uncle buys him one candy bar from those months, what is the probability Charlie will get a golden ticket?
One, or the movie would have been about someone else.