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Math Help - Dang Flies!!

  1. #1
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    Dang Flies!!

    Was wondering if I am reading to much into this problem.

    If two flies land on an orange, find the probability that they are on points that are within the same hemisphere

    I say the answer is 1, because there are 2 flies and 2 hemispheres 2/2=1..Is that a reasonable explanation or am I missing a bigger picture here?

    Thanks
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  2. #2
    Junior Member hoeltgman's Avatar
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    1 is probably wrong since it would mean that they are always on the same hemisphere, which is rather unprobable.

    Lets look from this point: The first fly lands on the orange. It doesn't really matter on which half. Now the second fly has two different possibilities. Either she lands on the same side or not.

    Therefore the probability is 0.5

    Alternatively write down all possibilities (should be 4) and those that suit (should be 2) 2/4 = 0.5
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  3. #3
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    Question

    So the fact that there are 2 flies and 2 hemispheres, makes it 4 possibilities?

    Tree Diagram

    F1 F2
    / \ / \
    H1 H2 H1 H2

    Appreciate the feedback
    Last edited by sarugaki; August 29th 2005 at 03:59 AM.
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  4. #4
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    The question as posed might be interpreted in two ways. If you mean "there's a hemisphere that they are both in", the answer is 1 (assuming that you intepret the exteme case of landing on antipodal points as included). If you mean "the same hemisphere with respect to some particular division of the orange into hemispheres (such as north and south)", the answer is 1/2.
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  5. #5
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    Given two flies at different points on the orange, there must exist a great circle that runs between the two flies. You can make the probability of them landing on different hemisphers 100% if you really want to .
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  6. #6
    hpe
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    Quote Originally Posted by sarugaki
    Was wondering if I am reading to much into this problem.
    If two flies land on an orange, find the probability that they are on points that are within the same hemisphere
    If the two flies land independently and the probability distributions for the landing sites both have densities wrt. Lebesgue measure on the sphere, the probability is indeed 1. You can always find a hemisphere that contains both flies, unless they land on antipodal points. This latter event has probability zero.
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  7. #7
    Junior Member hoeltgman's Avatar
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    The problem itself is rather tricky. It does not precise whether we are aloud to chose an hemisphere the way we want. In that case the probability is one or whether the hemispheres are fixed (as they are on Earth, North and South). Then the prob. is 0.5.
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