SURPRISE ITS A PARADOX, BUT I DON'T THINK SO
A teacher announced to his pupils that on exactly one of the days of the following
school week (Monday through Friday) he would give them a test. But it would be a
surprise test; on the evening before the test they would not know that the test would
take place the next day. One of the brighter students in the class then argued that the
teacher could never give them the test. "It can't be Friday," she said, "since in that
case we'll expect it on Thursday evening. But then it can't be Thursday, since having
already eliminated Friday we'll know Wednesday evening that it has to be Thursday.
And by similar reasoning we can also eliminate Wednesday, Tuesday, and Monday.
So there can't be a test!"
dbunker is a rankmaniac
The students were somewhat baffled by the situation. The teacher was well-known
to be truthful, so if he said there would be a test, then it was safe to assume that there
would be one. On the other hand, he also said that the test would be a surprise. But it
seemed that whenever he gave the test, it wouldn't be a surprise.
Well, the teacher gave the test on Tuesday, and, sure enough, the students were
surprised.
(excerpted from Joseph Halpern's Taken by Surprise: Surprise Test Revisited)
Example
there are 10 events of which 9 are false and 1 is true
i say first 9 are false = i say last 1 is true
in either case you are equally surprised by which one i chose
because you had no way of knowing which one was true
so
i say it didn't happen Monday to Thursday = i say it happened Friday
in either case you are equally surprised for the same reason.
the problem arises when you ask when it is you are surprised since in the
second case you have added an order of events. "on the evening before the test
you won't know the test will be the next day"
when people analyze this problem it is usually this sentence that is focussed
on, define evening: period from instant after potential-day-before-test
to instant before potential-today-test.
i argue that if you are surprised it didn't happen on potential-day-before-test
you must be characterized as surprised if it does happen on potential-today-test.
---------------------0 = ------------------------*
thus the fact that you are surprised it didn't happen on Monday to Thursday
= surprised it happened on Friday regardless of instant after potential-day-before-test,
instant before potential-today-test, or any time in between.
this turns this seemingly discreet problem into a continuous one, if you are told an event
can happen at any time in some time period you will be surprised even if it happens at the
last moment.
think of a game of chicken, people are surprised that they crashed because they are surprised
no one aborted before the crash.
0-------------------- = --------------0-------------- = ----------------------------0
the interesting thing about the last case is that it is the loss of the limited resource
(time without crash) that makes you surprised rather than the surprise itself, however,
they are still equivalent.
proof by contradiction
assume they are not surprised at the last moment
this implies that they are not surprised the second to last moment because they know the outcome
of the last moment (if last moment = crash then they don't crash now, if last moment = not crash
then they crash now) you can extend this argument to never being surprised, but they are surprised when it happens
this feels strange because we are essentially giving more meaning to the word surprised, surprised means
multiple events can happen, one happens, and you can always say this is true even in the case of
------------------------0, because no matter how close you get to 0, there is still a tiny - before 0.
so if you really asked the students "were you surprised the evening before class that the test was going
to be on Friday?", they either have to admit "I was surprised it didn't happen all the other times,
so yes i was surprised" because they game effectively ends on the potential-test point
on Thursday, everything after that is irrelevant.
So the problem with the girls induction is her base case of them not being surprised it happens on Friday.
what is really confusing in this problem is the separation of event evaluation from event updating. i.e.
you take the tests in the morning, but make evaluations in the evening, if you reread the question but say,
"at the moment i choose whether or not to give you the exam you will be surprised whether or not i give you the
exam the next day"
this is indisputably true on Friday because if he gives you the exam on Thursday you are surprised he doesn't give
it to you on Friday, if he does not give you the exam on Thursday, you are surprised he gives it to you on Friday.
so, the crucial thing to recognize is that the "time" between morning and evening is actually irrelevant. You can't
say you are any more or less surprised the evening than you were in the morning the instant
after he said the exam was or wasn't going to be on Thursday.
Enter this: hmrlx
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dave:
cmon people, this isn't that hard, just enter in your comment and the craptcha, then when it says unable to process, hit the back button, then refresh and enter in the restated captcha and there your comment is stored.
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douch:
<a href="http://www.rankcube.com/">rankmaniac</a>
Link: rankmaniac
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ypzhg:
<a href="http://scienceoftheweb.org/students/dbesse/rankmaniac.html">rankmaniac</a>
Link: rankmaniac
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info:
Here's a funny piece of info on crappy captchas: ratemyprofessors.com does institute a captcha when you want to start a new account. However, if you press the back button, you can try again on the same captcha and it will take it meaning with a brute force tactic can easily break break it. Sure this takes a long time, but that site is so broken i would feel bad attacking it (I'm also aware of the hypocrisy).
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tester:
stupid sessions on my stupid server are not working properly, so you may need to comment twice for it to work :s.
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Anon:
<a href="http://elainexmas.tumblr.com/post/54869378/eklee-is-a-rankmaniac">rankmaniac</a>
Link: rankmaniac
----------------------
Anon:
<a href="http://septosoft.com/rankmaniac">rankmaniac</a>
Link: rankmaniac
----------------------