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Math Help - Conditional Probability question

  1. #1
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    Conditional Probability question

    Bag M has 5 white balls and 2 red balls. Bag N contains 3 white balls and 4 red balls.
    If a white ball is selected from Bag N, What is the probability that a red ball was transferred from Bag M to Bag N?
    Please give the answer in any form you like, fractions, decimals, etc. Also, step by step instruction is also very much appreciated.
    Thank you in advance.
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  2. #2
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    Hello, flywithme!

    You left out part of the problem.
    I'll take a guess at what's missing.



    Bag M has 5 white balls and 2 red balls.
    Bag N contains 3 white balls and 4 red balls.

    A ball is randomly chosen from Bag M and moved to Bag N.

    If a white ball is selected from Bag N, what is the probability
    that a red ball was transferred from Bag M to Bag N?
    This is Conditional Probability, requiring Bayes' Theorem.

    . . P(\text{R moved }|\text{ W from N}) \;=\;\frac{P(\text{R moved }\wedge\text{ W from N})} {P(\text{W from N})} .[1]


    We consider the two cases . . .


    [1] White moved from Bag M: . P(\text{W moved}) \,=\,\frac{5}{7}
    . . Then bag N has 4W, 4R: . P(\text{W from N}) \,=\,\frac{4}{8}
    . . Then: . P(\text{W moved }\wedge\text{ W from N}) \;=\;\frac{2}{7}\cdot\frac{4}{8} \;=\;\frac{10}{28}


    [2] Red moved from Bad M: . P(\text{R moved}) \,=\,\frac{2}{7}
    . . Then Bag N has 3W, 5R: . P(\text{W from N}) \,=\,\frac{3}{8}
    . . Then: . P(\text{R moved }\wedge\text{ W from N}) \;=\;\frac{2}{7}\cdot\frac{3}{8} \;=\;\frac{3}{28} .[2]

    Hence: . P(\text{W from N}) \:=\:\frac{10}{28} + \frac{3}{28} \:=\:\frac{13}{28} .[3]


    Substitute [2] and [3] into [1]: . P(\text{R moved }|\text{W from N}) \;=\;\frac{\dfrac{3}{28}}{\dfrac{13}{28}} \;=\;\frac{3}{13}

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