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towr
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geography test
« on: Nov 19th, 2002, 12:19pm » |
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On a geography test you have to tell which of two german cities is greater in population for all possible pairs of the 80 largest cities of Germany. (And that's the only task on the test since it's allready 5 pages long)
But you didn't study last night, and only even recognize half the cities, and don't even know how those are ordered relative to each other..
Your friend on the other hand studied dutifully all night and recognizes all the cities and even knows how two cities are ranked relative to each other 60% of the time.
A week later you get the test-result and you have a higher score than your friend. How come?
hint: ignorance can be informative as well
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| « Last Edit: Nov 24th, 2002, 7:32am by towr » |
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TimMann
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Re: new: geography test
« Reply #1 on: Nov 19th, 2002, 5:06pm » |
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Nice. I didn't get it until I looked at the hint -- then I got it right away. My score was probably close to 75%.
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| « Last Edit: Nov 19th, 2002, 5:08pm by TimMann » |
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towr
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Re: new: geography test
« Reply #2 on: Nov 20th, 2002, 1:07am » |
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It will be slightly less than that (probably around 65%). I'm sure you'll guess why, so I won't explain and give away the complete answer to others just yet..
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Garzahd
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Re: new: geography test
« Reply #3 on: Nov 20th, 2002, 2:52pm » |
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In the 40% of the time that your friend doesn't know what's going on, does he answer randomly or incorrectly?
If incorrectly, then bingo, I agree with Tim. If randomly, then I might have my work cut out for me...
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Icarus
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Re: new: geography test
« Reply #4 on: Nov 20th, 2002, 3:37pm » |
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I don't see where you get 65%, towr. Because the information your hint refers to is not entirely reliable, I agree that Tim's number is on the high side. But I can't see any way to quantify how much, nor do I see any other factor that would decrease the expected score. Is there something I am missing?
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TimMann
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Re: new: geography test
« Reply #5 on: Nov 20th, 2002, 11:37pm » |
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Yes, my number was assuming that cities I've heard of are bigger than cities I haven't heard of. That's an excellent guess, but won't always be right. However, I don't see any way to estimate how often it will be wrong. It just depends on how quirky my knowledge of German cities happens to be. If by good luck the cities I've heard of are all larger than the ones I haven't, then about 50% of the questions will compare a city I've heard of with one I haven't, and I'll get those right. On the questions comparing two cities both of which or neither of which I've heard of, I'll get about half right, for a total of 75% right.
In real life I might do even better, perhaps more than compensating for not knowing about some large cities, because I probably have some idea about the relative size of the cities I've heard of. Like, I'm sure Berlin is very large, so if I say it's larger when comparing it with another city I've heard of instead of guessing randomly, I should collect some additional right answers.
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towr
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Re: new: geography test
« Reply #6 on: Nov 21st, 2002, 12:51am » |
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Garzahd: If they have no way of deducing the answer they'll gamble with 50% accuracy.
Icarus: yes, the information isn't totally reliable, that is why the score isn't 75%. But in so far as this problem is based on an actual experiment I know the reliability is about 80%. Of course unless you've read the same book ("simple heuristics that make us smart") or a related article I wouldn't expect you to know that number (it's an average anyway, so it will vary per individual).
But intuitively you'd expect a known city to be larger than one you've never heard of, or "if I haven't heard of it, it can't be that important". Which is the answer to the problem (the score doesn't really matter)
In real life you'll always have extra information, you can for instance use familiarity of a name as a measure of it's size "I've read more about Berlin than Hannover, so Berlin is probably larger". And that will improve the score in the 'known cities'-pairs.
Aside from that you may know Berlin is a capital, and thus probably larger than any other german city (100% ecological validity, but only applicable in 1/79 of the time) Other good cues are 'exposition site' (91%), 'soccer team' (87%) etc
These are easier to calculate than recognition validity, since you can find them out in the world. To get an approximation of the recognition validity you can check how often a city get's mentioned in the media (f.i. a newspaper, since that is easiest to check on), and take the 50% most often mentioned as the 50% that will be recognized, then calculate the validity.
As for further empirical evidence. They did the same test with German and American students, both for German and American cities. The Germans scored higher on American cities, and the Americans higher on German cities.
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| « Last Edit: Nov 21st, 2002, 12:54am by towr » |
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Icarus
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Re: new: geography test
« Reply #7 on: Nov 21st, 2002, 10:00am » |
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Thanks. When I saw your 65%, I thought that perhaps there was something specific that I was missing that allowed you to calculate it. Since it was based on information I didn't have, I can relax. 
Garzahd has a point. If your friend knows 60% of the answers and guesses randomly on the other 40%, then he can expect a score of ~80%, far greater than your 65%. On the basis of this, I assumed that the problem meant the friend will answer the other 40% wrong. (He must not of studied that hard if he can be random chance by only 10%  .)
This has been an interesting problem. Thanks for posting it.
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towr
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Re: new: geography test
« Reply #8 on: Nov 21st, 2002, 1:06pm » |
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euhm.. yes..
Well, it's somewhat the difference between knowing, and thinking you know.. He studied hard, so he thinks he knows the answers pretty well, in actuallity he only knows (has the correct belief) 60% of the time. And he doesn't know which he knows, and which he only thinks he knows :p
Learning the ranks of 80 cities isn't that easy to begin with.. Especially if you don't have a good strategy. (Besides studying all night long is a bad idea, because your memory doesn't work as well without a good night's sleep)
Anyway, the point is he gets 60% right.. Though I should have been more clear on that..
In the book the idea is that if they know both cities they'll use the knowledge about the cities (cues like 'does the city have a soccer-team') to deduce which is larger. But sometimes that information won't point in the right direction.
And the person that only recognizes some of the cities would still use his knowledge on the cities he does know (an aspect I left out here)
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TimMann
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Re: new: geography test
« Reply #9 on: Nov 21st, 2002, 2:39pm » |
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 I'd probably do worse at guessing which German cities have soccer teams than guessing which ones are larger.
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Kozo Morimoto
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Re: new: geography test
« Reply #10 on: Nov 24th, 2002, 6:24am » |
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I'm still not getting this...
for all possible pairs of 80 cities would make 3159 pairs?
And there are 3 possibilities:
a) you know both cities in the pair
b) you know one of the cities in the pair
c) you know neither of the cities in the pair
So if you know 40 cities (but not their rank), you would have 779 pairs in (a), 779 pairs in (c), 1601 pairs in (b)?
You are saying you'd get 50% right by gueesing in (c)?
And what are the empirical odds for (a) and (b) ?
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towr
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Re: new: geography test
« Reply #11 on: Nov 24th, 2002, 7:31am » |
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You can make an eductated guess based on the recogniztion-heuristic when you know one of the two cities. Which is the case for 40*40 = 1600 pairs (just over half).
In this case the validity is about 80%, so you'd get about 1280 right.
If you guess at chance level (50%) for the other (40*39) 1560 pairs (where you know either both, or neither of the cities), you'll score 780 there.
So in total the score is 2060 out of 3160 = 65.2%
In general if you have strategies for each case:
a) you know both cities in the pair
b) you know one of the cities in the pair
c) you know neither of the cities in the pair
with validities of a,b,c respectively,
your score with N cities of which n are recognized is
a * (N-n)/N * (N - n -1)/(N-1) +
b * 2*n/N*(N-n)/(N-1) +
c * (n)/N * (n -1)/(N-1)
As long as b is larger than a it will often be advantageous not to know every city (also known as the less-is-more effect)
(You can easily find the optimum in the curve once a,b and c are known. As long as a,b and c don't depend on n.. Which of course they do in real life :p)
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P.J.
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I call shenanigans on this riddle.
The solution expects knowledge of obscure heuristic stats, and unreliable information? Plus using heuristics only for the "unknown vs. known," and not for the "known vs. known" seems inconsistant. 80% when it's known vs. known, but only 50% when it's (Berlin vs Potsdam) or (Munich vs. Heidelberg)
I've enjoyed working through the riddles here, I just think this one needs reworded.
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towr
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Re: geography test
« Reply #13 on: Nov 11th, 2003, 6:33am » |
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I disagree. And the actual percentages is not important.
Virtually everyone knows that the names of cities they know (disregarding those for which they have other obvious reasons for knowing them, such as living in them) are those of the more important cities , and that these are usually larger.
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| « Last Edit: Nov 11th, 2003, 7:02am by towr » |
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Hmm, I was going way out there to solve this.
I figured the answer was either I guessed really really well.
OR
Since There was a week between the test and the results, I would have a week to relocate millions of German citizens (by gunpoint if necessary) into the various cities in the order that I had ranked them on the test.
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