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Suppose that there is one hotel with nine floors (first floor = ground floor + 1) where the math seminar takes place, four brilliant mathematicians who are guests of the hotel, one drunk receptionist and four invitation letters for night party each addressed to single one of the mathematicians. All mathematicians are in the conference room which is on the ground floor . At the one moment one of the mathematicians comes out of the conference room and goes to the floor where his room is by elevator. Since he is drunk, receptionist forgets to deliver a invitation letter to mathematician. Ten minutes later, since he is drunk, he takes randomly one of the letters instead of all four and goes after mathematician. Receptionist enters the elevator but he doesn't know on which floor mathematician's room is. Only thing that he knows for sure is that room isn't on second, fourth, sixth and ninth floor .

What is probability that receptionist on the first try goes to the right floor where the mathematician is and that letter which he carries is addressed exactly to that mathematician?

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    Seems like it should be 1/20 but this question is too twisty to make sense of. Isn't it possible the drunk receptionist will enter and then immediately exit the elevator and then attempt to find the mathematician on the first floor where the reception also presumably is? What floor is the conference room on and does it matter? Not asking for spoilers of course. :)2011-09-15
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    @Dan,Conference room is on the ground floor2011-09-15
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    This had such an alluring title, I'm a little disappointed in the delivery. :)2011-09-15
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    Would the answer be any different if the mathematicians weren't so brilliant?2011-09-15
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    Why does the question have so many floors if some of them are irrelevant? Or are they not irrelevant because the receptionist might be wrong or picks a floor uniformly at random including the floors he knows are wrong?2011-09-15
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    @PeterTaylor,Those floors are irrelevant...and receptionists would not make such mistake2011-09-15
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    @pedja, so why complicate the question by adding them?2011-09-15
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    @Peter,there is a reason for such formulation...it is true that question may be formulated in the other way also2011-09-15
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    This question is entirely missing motivation: where does it come from? Why do you want to know the answer? What makes this a question of *general interest* rather than something localized to a very particular (rather contrived) situation?2011-09-15
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    @PeteL.Clark,number of floors=$\frac{n}{2}-1$ ,number of mathematicians=number of primes less than $\frac{n}{2}$ ,number of allowed floors= (number of odd numbers less than $\frac{n}{2}$)-1+1 ,number of addressed letters=number of primes between $\frac{n}{2}$ and $n$2011-09-15
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    @pedja: If you are actually interested in a number theoretic question, why are you asking only about a single value of $n$ and phrasing it in terms of a hotel?2011-09-15
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    @PeteL.Clark,I tried to make an interesting model in order to examine probability of appearance of such prime pairs which sum is equal to $n$ , but it seems that I have failed.2011-09-15
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    @pedja: in general, turning a math question into a "story problem" tends to obscure the underlying ideas, not bring them out. (There are exceptions, even in this genre of story problem: "Hilbert's hotel" is a decently useful metaphor for thinking about countably infinite sets.) If I am trying to maximize the area enclosable by a fence of given perimeter, it does not help me to hear that the fence is being built by a farmer named Octavia. I honestly can't tell whether you are just playing around here. If not, you should probably ask your question again in "naked number theoretic" form.2011-09-15
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    @PeteL.Clark,http://math.stackexchange.com/q/62886/156602011-09-16
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    @Pete, I don't agree that that story problems tend to obscure the details. Doesn't Octavia build octagonal fences? What is an "obscuration" if not a "problem"?2011-09-30
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    @Dan,"The mathematics are usually considered as being the very antipodes of Poesy. Yet Mathesis and Poesy are of the closest kindred, for they are both works of the imagination." -Thomas Hill2011-09-30

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Presumably the receptionist could have gone to one of the floors 3, 5, 7, or 8 with equal probability (though if he's so plastered he didn't bother checking which letter he picked, one wonders how he pushed a button corresponding to one of the intended floors). And presumably the mathematician is on one of these floors with equal probability. Then the problem of being on the same floor is equivalent to asking: if two people each pick a number from $\{3,5,7,8\}$, what's the probability they are the same? There's $4$ ways they could each pick the same number, and $4\times4=16$ ways they could each pick some number independently, so there is a $4/16=1/4$ chance of collision. Now the chance the letter is the correct one is $1/4$, and completely independent of whether or not the receptionist got the right floor, so the chance that the receptionist got both the floor correct and the letter correct is $1/4\times1/4=1/16$.


Alternatively, if both the mathematician and the receptionist may end up on the first floor, the numbers are instead $\{1,3,5,7,8\}$, giving a final probability of $\frac{5}{5\times5}\times1/4=1/20$ by our previous reasoning. If the mathematician's floor may be the first but the receptionist is destined to go up, then for every admissible floor there is a $1/5$ chance the mathematician winds up there, and for every nonfirst admissible floor there is a $1/4$ chance of the receptionist winding up there, so the chance they collide is $(1/5)\times(1/4)$ times four (one for each nonfirst admissible room), or $1/5$. The letter is again a $1/4$ chance of being correct, and independent of floor, so the chance the floor and letter are correct is $1/5\times1/4=1/20$.

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    ,You omitted first floor in your calculation.Mathematician could be there also2011-09-15
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    pedja, the mathematician isn't drunk, so he wouldn't take the elevator to the same floor he is already on. Oh, but maybe the receptionist would! :)2011-09-15
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    @pedja: If the mathematician was sober and had to go up the elevator to get to her floor, then she couldn't have been on the first floor: that would imply she acted illogically, a contradiction. :) [If you clarify the meaning of the OP, I might edit; be sure to mention whether or not the receptionist might potentially *not* use the elevator.]2011-09-15
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    @anon,According to the text of the question mathematician's room can be on the first floor2011-09-15
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    @pedja: Then either the text is positing a "*brilliant*" mathematician being illogical (blasphemy!) or they didn't say the mathematician used the elevator. Once you answer my other question - might the receptionist also check the first floor? - I will edit.2011-09-15
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    @anon,Receptionist doesn't know whether mathematician is on the first floor,but he knows that he could be there also2011-09-15
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    @Dan,As far as I know first floor = ground floor +12011-09-15
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    pedja, then is the ninth floor you mentioned on the roof? :) In the U.S., "first floor" and "ground floor" mean the same thing.2011-09-15
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    @anon,First floor = ground floor +1 ,and reception is on the ground floor2011-09-15
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    @pedja: In the US ground floor = first floor, so that confused me.2011-09-15
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    @Dan,it seems that I should replace words "ground floor" with basement2011-09-15
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    @anon,So, I should write "basement" instead ground floor?2011-09-15
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    @pedja: No, in the US basement = ground floor - 1, same as elsewhere. Just mention the convention that first floor = ground floor + 1 in the question and you should be fine.2011-09-15
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    pedja, basements are underground, which is a very unattractive location for a hotel lobby and conference room.2011-09-15
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    @Dan, I used to live in a building in New York City where the entrance was called L for "lobby" and then the floor *above* that was the basement. Then came 1, 2, etc.2011-09-15
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    I know of at least one building in the USA where the floor officially called the ground floor is the one below the floor officially called the first floor, and the basement is below the ground floor (and the second, third, and fourth floors are above the first).2011-09-16