At least in the boat problem one can give an interval estimate:)
Seriously, though, this is interesting stuff. I thought about writing a post on 'different assumptions' like this, like when you have explained to a student how to calculate the molecular mass of a compound, show them the very rudiments of mass spectrometry theory and watch them try to reconcile the idea of a structured molecule (or whatever it is called in salts) with the idea of just totalling the nuclons and dividing by the charge. I had problems with it when I had to read about it.
Or there's another case of diatoms' shells inheritance, compared to Mendelian genetics.
If anyone has an idea of another 'similar-but-not-similar' pair of problems, please let me/us know.
I think the issue here might be slightly different than posed. I think the real issue is that children instinctively assume they're running on corrupted hardware. For all priors in math, they've had a solvable problem. They've had problems they couldn't solve, and then been shown it was a mistake on their part. Without good cause, why would they suddenly assume all their priors are wrong, and not just that they're failing to grasp it? Given their priors and information, it's ration to expect that they missed something.
Yes, I agree. It shows children are trying to guess the teacher's password and are not doing math. Interestingly, when I asked my son this question he said you couldn't find the answer. When I asked how he knew that he said he has seen other math problems where you don't have enough information to solve.
You are a bus driver and you start with an empty bus. On the first stop 7 people got in. On the second stop 4 more people got in and two people left. On the third stop no one got in and one person left. On the fourth stop 5 people got in, 2 left. On the fifth stop one got in and two left. What is the colour of the driver's eyes?
The prior is that his eyes are brown, since most people have brown eyes. But in the most populous countries with brown eyes and busses, the busses always have way more than 7 people on them, so that slightly shifts the probability towards non-brown colors. But even in countries where there are fewer bus passengers, most people have brown eyes, so brown is still most likely.
I wonder how my coworkers will do...
EDIT (2016.10.21): In case anybody is interested, the results with my coworkers are...
6 variations of "I don't know" with one outright "You didn't give me any information about the shepherd... he could be any age"
4 numeric answers ranging from 5 to 35
1 got distracted and never answered the question
I've got a party to attend tomorrow, we'll see if they do better.
At least in the boat problem one can give an interval estimate:)
Seriously, though, this is interesting stuff. I thought about writing a post on 'different assumptions' like this, like when you have explained to a student how to calculate the molecular mass of a compound, show them the very rudiments of mass spectrometry theory and watch them try to reconcile the idea of a structured molecule (or whatever it is called in salts) with the idea of just totalling the nuclons and dividing by the charge. I had problems with it when I had to read about it.
Or there's another case of diatoms' shells inheritance, compared to Mendelian genetics.
If anyone has an idea of another 'similar-but-not-similar' pair of problems, please let me/us know.
I think the issue here might be slightly different than posed. I think the real issue is that children instinctively assume they're running on corrupted hardware. For all priors in math, they've had a solvable problem. They've had problems they couldn't solve, and then been shown it was a mistake on their part. Without good cause, why would they suddenly assume all their priors are wrong, and not just that they're failing to grasp it? Given their priors and information, it's ration to expect that they missed something.
Yes, I agree. It shows children are trying to guess the teacher's password and are not doing math. Interestingly, when I asked my son this question he said you couldn't find the answer. When I asked how he knew that he said he has seen other math problems where you don't have enough information to solve.
Reminds me of a slightly different problem:
You are a bus driver and you start with an empty bus. On the first stop 7 people got in. On the second stop 4 more people got in and two people left. On the third stop no one got in and one person left. On the fourth stop 5 people got in, 2 left. On the fifth stop one got in and two left. What is the colour of the driver's eyes?
The prior is that his eyes are brown, since most people have brown eyes. But in the most populous countries with brown eyes and busses, the busses always have way more than 7 people on them, so that slightly shifts the probability towards non-brown colors. But even in countries where there are fewer bus passengers, most people have brown eyes, so brown is still most likely.
Read the first sentence of the problem again.
I didn't get it until I read your comment.
This reminds me of this comment of mine, although it is not directly related.
9eB1, what color are your eyes (if you don't mind me asking)?
I'm not sure, I have different forms of identification that state that they are different colors, but they are not brown.
Do you have a mirror?