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Math Help - a little probability question

  1. #1
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    a little probability question

    An elevator goes from the ground floor up to the 8th floor (8 floors in total). In the elevator 10 people. Each one of them choose randomly and independently with the other people in which floor to get off the elevator. The expected value (mean) of the number floors in which no one came off at is:

    a. 6 b. 8 c. 8*(7/8)^10 d. 10*(1/8)^10 e. 10*(7/8)^8

    the correct answer should be "e". why ?

    cheers
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  2. #2
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    The probability that any PARTICULAR X floors are not used, is 1 - (\frac{X}{8})^{10} =

    As there are \binom{8}{X} ways to choose X floors, it is clear that the probability of X floors not being chosen is just \binom{8}{X} \cdot (1 - (\frac{X}{8})^{10})

    Hence just sum the above for X in 1-8 and you should get the correct answer.
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  3. #3
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    I am afraid I don't understand that first line of the solution, can you explain is please ? thnx
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  4. #4
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    I think the easiest way to answer this question is using reasoning like in this post. My grasp of this reasoning is not quite as good as it should be, so I won't offer to explain, but I believe it applies.

    Then again, I might be completely off-base.
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  5. #5
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    let's make it easier...
    I need E(X). So, can someone write to me what is the probability P(X=0), the probability that 0 floors were not used, and P(X=1), I'll take it from there.
    thanks
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  6. #6
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    Yes that solution is much nicer, thanks.

    Following its construction, define the random variable X_i such that

    X_i = 1 if floor i isn't used and X_i = 0 if floor i is used.

    Clearly then you want to calculate E(\sum_{i=1}^{8} X_i) = \sum_{i=1}^{8} E(X_i) = 8 \cdot E(X_1)

    (Do you see why?).
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  7. #7
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    no, I am afraid not...sorry. actually I think your first way was clearer to me, can you go back to it and tell me what P(X=0) is equal to ?
    I need to get 10*(7/8)^8. can't see how to get it from the second method...
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  8. #8
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    You don't need to know P(X=0) because it contributes nothing to E(x).
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  9. #9
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    I know, I just thought that if I see an example of how you calculate P(X=0) I can calculate for the rest of X=1-8 and then calculate E(X). I said that 'e' is the correct answer, but I am starting to wonder, according to what you guys said, that it's not true, that 'c' is the correct answer ! am I right ?
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  10. #10
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    Yes "c" is the answer.

    Can you figure out why the probability that any given door not being chosen is (\frac{7}{8})^{10}?

    Hint: First ask yourself what is the probability of a given person NOT choosing this floor? Then use this to determine the probability that all 10 of them don't choose it.

    I already described in my first post how to calculate each P(x=i) for i in 1-8, but this is long and unnecessary.
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  11. #11
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    got it !
    thank you !!! :-)
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