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1 vote (11.11%)
4 votes (44.44%)
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3 votes (33.33%)
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1 vote (11.11%)

9 members have voted

endermike
endermike
Joined: Dec 10, 2013
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May 6th, 2014 at 10:25:22 PM permalink
I don't remember how to find the solution off the top of my head and I plan on going to bed shortly.
Wizard
Administrator
Wizard
Joined: Oct 14, 2009
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May 6th, 2014 at 10:27:59 PM permalink
Quote: sodawater

How can this be right, though? He didn't specify how many cases. If the game were just two cases, you could beat that by blindly picking one.



My answer is not appropriate for small numbers of cases, like less than ten. You specifically do not know anything about the distribution. However, the way this problem is usually stated is every suitcase has a different amount, and you're given this information before starting.
It's not whether you win or lose; it's whether or not you had a good bet.
sodawater
sodawater
Joined: May 14, 2012
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May 6th, 2014 at 10:29:21 PM permalink
Quote: ThatDonGuy

This is the Secretaries (aka Sultan's Dowry) problem, for which the answer is something like, pass on the first N/e (e as in the base of natural logarithms) briefcases, then take the first one you open that is higher than the rest that you had opened up to that point; the probability is slightly higher than 1/e, so, given the choices, I'd say "around 1/3".



Very interesting.

http://en.wikipedia.org/wiki/Secretary_problem

OP did specify the minimum probability of success following optimal strategy, so 1/e looks correct. But in the exact case of 2 cases, you could achieve 1/2 by any strategy. However 1/e is less than 1/2 so I guess it technically is still correct.
sodawater
sodawater
Joined: May 14, 2012
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May 6th, 2014 at 10:35:09 PM permalink
Quote: Wizard

My answer is not appropriate for small numbers of cases, like less than ten. You specifically do not know anything about the distribution. However, the way this problem is usually stated is every suitcase has a different amount, and you're given this information before starting.



From wikipedia:




So it looks like the wording of the original problem (and the answer choices) could have been modified a little to avoid these cases.
chrisr
chrisr
Joined: Dec 9, 2013
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May 7th, 2014 at 8:41:03 AM permalink
I didn't know this problem was so well known. I guess it makes sense since the final answer is quite simple.

The probability of winning is always greater than 1/e regardless of N, so around 1/3 is the best answer.


A harder problem: You know the distribution of the cases is uniformly distributed between $0 and $1,000,000. What is the probability of winning now? [split thread]

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