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Math Help - Ball Probability question

  1. #1
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    Ball Probability question

    Suppose that we have a white urn containing six balls and one red ball and we have a red urn containing one white ball and six red balls. An experiment consists of selecting at random a ball from the white urn and then (without replacing the first ball) selecting at random a ball from the urn having the color of the first ball. Find the probability that the second ball is red.

    The probability the second ball is red is?
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  2. #2
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    Re: Ball Probability question

    Quote Originally Posted by randyeric View Post
    Suppose that we have a white urn containing six balls and one red ball and we have a red urn containing one white ball and six red balls. An experiment consists of selecting at random a ball from the white urn and then (without replacing the first ball) selecting at random a ball from the urn having the color of the first ball. Find the probability that the second ball is red.

    The probability the second ball is red is?
    There are two event chains that lead to the second ball selected being red.

    (select W, select R)
    (select R, select R)

    let's look at probabilities of the first ball selection.

    $Pr[W]=\dfrac 6 7$

    $Pr[R]=\dfrac 1 7$

    Now

    $Pr[R|W]=\dfrac 1 6$ as the white urn contains (5 white, 1 red) after having 1 white ball removed.

    $Pr[R|R] = \dfrac 6 7$ as it has been untouched

    So our total probability

    $Pr[R \mbox{ is second selection}]=Pr[R|W]Pr[W] + Pr[R|R]Pr[R] = \dfrac 1 6 \dfrac 6 7 + \dfrac 6 7 \dfrac 1 7 = \dfrac {13}{49}$
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