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? regarding probability

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  • travelguy027
    I am stuck on a probability question .... Can anyone help ? =) The Question is ... you have 2 boxes ... one contains 1 white marble and 2 black marbles the
    Message 1 of 4 , Dec 1, 2003
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      I am stuck on a probability question ....

      Can anyone help ? =)

      The Question is ...

      you have 2 boxes ...
      one contains 1 white marble and 2 black marbles
      the other box contains 3 black marbles and 4 white marbles
      without looking at the color of the marble - 1 marble is randomly
      drawn from one of the boxes (also randomly selected) and placed into
      the remaining box. If you now draw a marble out of the box to which
      the marble was added ... what is the probability that it is white ?

      My best guess after considerable work is 157/336

      I hope someone can confirm this - or educate me about where I went
      wrong ! thanks
      Mark
    • Noel Vaillant
      ... Mike, I agree with 157/336. The challenge is to come up with a clear and convincing proof... I am gonna think about it. Noel.
      Message 2 of 4 , Dec 1, 2003
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        > My best guess after considerable work is 157/336

        Mike,

        I agree with 157/336. The challenge is to come up with a clear and
        convincing proof... I am gonna think about it.

        Noel.
      • Pedro Tytgat
        Hi, using a three-level probability tree, I find 349/672... Pedro ... Van: travelguy027 [mailto:travelguy027@aol.com] Verzonden: maandag 1 december 2003 18:15
        Message 3 of 4 , Dec 1, 2003
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          Hi,

          using a three-level probability tree, I find 349/672...

          Pedro

          -----Oorspronkelijk bericht-----
          Van: travelguy027 [mailto:travelguy027@...]
          Verzonden: maandag 1 december 2003 18:15
          Aan: probability@yahoogroups.com
          Onderwerp: [probability] ? regarding probability


          I am stuck on a probability question ....

          Can anyone help ? =)

          The Question is ...

          you have 2 boxes ...
          one contains 1 white marble and 2 black marbles
          the other box contains 3 black marbles and 4 white marbles
          without looking at the color of the marble - 1 marble is randomly
          drawn from one of the boxes (also randomly selected) and placed into
          the remaining box. If you now draw a marble out of the box to which
          the marble was added ... what is the probability that it is white ?

          My best guess after considerable work is 157/336

          I hope someone can confirm this - or educate me about where I went
          wrong ! thanks
          Mark




          [Non-text portions of this message have been removed]
        • Noel Vaillant
          ... into ... which ... Mike, I am not too happy with what follows, as I don t find it very elegant, but I cannot think of anything better for the time being.
          Message 4 of 4 , Dec 2, 2003
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            > you have 2 boxes ...
            > one contains 1 white marble and 2 black marbles
            > the other box contains 3 black marbles and 4 white marbles
            > without looking at the color of the marble - 1 marble is randomly
            > drawn from one of the boxes (also randomly selected) and placed
            into
            > the remaining box. If you now draw a marble out of the box to
            which
            > the marble was added ... what is the probability that it is white ?
            >
            > My best guess after considerable work is 157/336

            Mike,

            I am not too happy with what follows, as I don't find it very
            elegant, but I cannot think of anything better for the time being.

            Let X0 be a radom variable with values in {0,1}, modelling the draw
            of one marble in box 0, i.e. with distribution:

            P(X0=0)= 1/3 and P(X0=1)=2/3

            where it is understood that '0' means 'white' and '1' means 'black'.
            Likewise, let X1 be a random variable with values in {0,1}, modelling
            the draw of one marble in box 1, i.e. with distribution:

            P(X1=0)=4/7 and P(X1=1)=3/7

            Let e be a random variable with values in {0,1}, modelling the radom
            choice between box 0 and box 1, i.e. with distribution:

            P(e=0)=1/2 and P(e=1)=1/2

            where it is understood that 'e=0' means that box 0 has been selected,
            whereas 'e=1' means that box 1 has been selected.

            One marble will be drawn from one box and added to the other. If box
            0 is initially selected (e=0), then box 1 will contain 8 marbles. If
            box 1 is initially selected then box 0 will contain 4 marbles.

            Let e0 be a random variable with values in {0,1} modelling whether a
            draw from "box 0 with one additional marble" will result in a draw of
            a marble from box 0, or a draw of the additional marble, i.e. with
            distribution:

            P(e0=0)=3/4 and P(e0=1)=1/4

            where it is understood that if e0=0 then the draw from "box 0 with
            one additional marble" brings a marble from box 0, whereas e0=1 means
            that the additonal marble has been drawn.

            Let e1 be a random variable with values in {0,1} modelling whether a
            draw from "box 1 with one additional marble" will result in a draw of
            a marble from box 1, or a draw of the additional marble, i.e. with
            distribution:

            P(e1=0)=7/8 and P(e0=1)=1/8

            where it is understood that if e1=0 then the draw from "box 1 with
            one additional marble" brings a marble from box 1, whereas e1=1 means
            that the additonal marble has been drawn.


            Let X be the radom variable modeling our experiment (i.e. the draw
            from the second box). Then X is defined as:

            If e=0 and e1=0 then X=X1
            If e=0 and e1=1 then X=X0

            If e=1 and e0=0 then X=X0
            If e=1 and e0=1 then X=X1


            Choices in box 0 and 1 (i.e. X0 and X1) , choice between box 0 and
            box 1 (i.e. e), choices wether additional marble or initial marbles
            (i.e. e0 and e1), should all be independent in our model, i.e.

            X0,X1,e,e0,e1 are independent.

            Our question is to compute P(X=0). This goes as follows:

            P(X=0)

            =P(e=0,e1=0,X=0)
            +P(e=0,e1=1,X=0)
            +P(e=1,e0=0,X=0)
            +P(e=1,e0=1,X=0)

            =P(e=0,e1=0,X1=0)
            +P(e=0,e1=1,X0=0)
            +P(e=1,e0=0,X0=0)
            +P(e=1,e0=1,X1=0)

            =P(e=0)P(e1=0)P(X1=0)
            +P(e=0)P(e1=1)P(X0=0)
            +P(e=1)P(e0=0)P(X0=0)
            +P(e=1)P(e0=1)P(X1=0)

            =(1/2)(7/8)(4/7)
            +(1/2)(1/8)(1/3)
            +(1/2)(3/4)(1/3)
            +(1/2)(1/4)(4/7)

            =(1/2)(1/8)[4+(1/3)+2+(8/7)]
            =(1/2)(1/8)(1/3)(1/7)[84+7+42+24]
            =157/336

            Noel.
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