The bat and ball problem I answer in what I’ll call one conscious time-step with the correct “five cents”, but it happens too fast for me to verify how (beyond the usual trouble with verifying internal reflection). I would speculate, in decreasing order of intuitive probability, that in order to get the answer, either (a) I’ve seen an exactly analogous “trick” problem before and am pattern-matching on that or (b) I’m doing the algebra quickly using my seemingly well-developed mathematical intuition. I can also imagine (c) I’m leaping to the “wrong” answer, then trying to verify it, noticing it’s wrong, and correcting it, all in the same subconscious flash, but that feels off. Imagining the “ten cents” answer doesn’t actually feel compelling; it just feels wrong. (It feels like a similar emotion to noticing I’ve gotten the wrong amount of change, in fact.)
The widgets problem I do a noticeable double-take on, but it’s rapidly corrected within one conscious time-step; the “100” is a momentary flicker before my brain settles on the correct answer. Imagining “100” afterwards feels wrong, but less immediately so than “ten cents” did. It feels like I have a bias there toward answering “how many widgets can you produce in a fixed time” questions, so I might have an echo of the misreading “how many widgets can 100 machines produce in [assumed to be the same amount of time as before, since no contrary time value is presented to override this]”.
The lily pads question takes me a conscious time-step longer to answer than either of the other two; the initial flash is “inconclusive”, and then I see myself rechecking the part where the quantity doubles every step before answering “47”. (I notice I didn’t remember that the steps were days, only remembering that there was a time unit; I don’t know if that’s relevant.) Imagining “24” afterwards feels some intermediate level of wrong between “ten cents” and “100”; my mental graph of the growth curve puts the expected value 24 at “way too low” intuitively before I can compute the actual exponent.
The bat and ball problem I answer in what I’ll call one conscious time-step with the correct “five cents”, but it happens too fast for me to verify how (beyond the usual trouble with verifying internal reflection). I would speculate, in decreasing order of intuitive probability, that in order to get the answer, either (a) I’ve seen an exactly analogous “trick” problem before and am pattern-matching on that or (b) I’m doing the algebra quickly using my seemingly well-developed mathematical intuition. I can also imagine (c) I’m leaping to the “wrong” answer, then trying to verify it, noticing it’s wrong, and correcting it, all in the same subconscious flash, but that feels off. Imagining the “ten cents” answer doesn’t actually feel compelling; it just feels wrong. (It feels like a similar emotion to noticing I’ve gotten the wrong amount of change, in fact.)
The widgets problem I do a noticeable double-take on, but it’s rapidly corrected within one conscious time-step; the “100” is a momentary flicker before my brain settles on the correct answer. Imagining “100” afterwards feels wrong, but less immediately so than “ten cents” did. It feels like I have a bias there toward answering “how many widgets can you produce in a fixed time” questions, so I might have an echo of the misreading “how many widgets can 100 machines produce in [assumed to be the same amount of time as before, since no contrary time value is presented to override this]”.
The lily pads question takes me a conscious time-step longer to answer than either of the other two; the initial flash is “inconclusive”, and then I see myself rechecking the part where the quantity doubles every step before answering “47”. (I notice I didn’t remember that the steps were days, only remembering that there was a time unit; I don’t know if that’s relevant.) Imagining “24” afterwards feels some intermediate level of wrong between “ten cents” and “100”; my mental graph of the growth curve puts the expected value 24 at “way too low” intuitively before I can compute the actual exponent.