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Probability question: help

  1. Sep 22, 2006 #1
    that day my friend asked me a question. what is the probability that the chord inside a circle (with radius 1) will be greater than D, where D is in the interval [0, 2].

    i have came up with the answer
    [tex]{\pi - 2\sin^{-1}(D/2)} \over {\pi}[/tex]

    is it ok? or did i do anything wrong?
     
    Last edited: Sep 23, 2006
  2. jcsd
  3. Sep 22, 2006 #2

    HallsofIvy

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    Am I missing something? A circle of radius 1 has diameter of length 2 and all other chords are shorter. It is impossible for a chord to have length greater than 2. The probability is 0.
     
  4. Sep 22, 2006 #3

    mathman

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    If you put D=2 in the expression, you will get 0. The question is what is the probability for a specific D, where 0<D<2. The answer will depend on what sort of distribution function defines the placing of the chord - there is more than 1 way to do it.
    Examples:
    (1) Uniform in distance from the center.
    (2) Pick a point on the circumference, other end point is uniform around the circumference.
     
  5. Sep 23, 2006 #4
    how will the 2 distributions affect the answer? can you be a bit more elaborate?
     
  6. Sep 23, 2006 #5

    HallsofIvy

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    Ah, yes, I simply misread the quesition!
     
  7. Sep 23, 2006 #6

    mathman

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    You had to make some sort of assumption about the chords to get the answer you did. I haven't worked out what the rersults would be for the 2 examples I gave, but I can make up possibilities which I know would give different results, although they might look strange. For example, uniform in the square of the distance from the center.
     
  8. Sep 24, 2006 #7

    mathman

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    I took the time to work out the various possibilities I mentioned. To simplify notation, let s=D/2. The probabiliites for these cases are:

    Case.......................................................Prob.
    uniform in arc length.............................murshid islam result
    uniform in distance from center..............(1-s2)1/2
    uniform in distance squared...................1-s2
    uniform in chord length.........................1-s

    As you can see, there is no "right" answer.
     
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