Each of three bags A, B, C contains white balls and black balls. A has a1 white & b1 black, B has a2 white & b2 black and C has a3 w

Question

Each of three bags A, B, C contains white balls and black balls. A has a1 white & b1 black, B has a2 white & b2 black and C has a3 white & b3 black. A ball is drawn at random and is found to be white. Find the respective probability that it is from A, B & C.

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Athena 3 weeks 2022-01-06T23:21:46+00:00 1 Answer 0 views 0

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    2022-01-06T23:23:42+00:00

    Answer:

    See explanation ( Answers are too long)

    Step-by-step explanation:

    We will first compute a general probability for picking a white ball:

              P (W) = a_1 / (a_1 + b_1) + a_2 / (a_2 + b_2) + a_3 / (a_3 + b_3)

    part a)

    We are asked to find the probability of white ball given that it pulled from bag A. So if we express it in notation we are asked for P ( A / W). We will use conditional probability to answer our question:

                               P ( A / W ) = P ( W & A ) / P (W)

                               P ( W & A ) = a_1 / (a_1 + b_1)

    Hence,

    P ( A / W ) = [a_1 / (a_1 + b_1)] / [a_1 / (a_1 + b_1) + a_2 / (a_2 + b_2) + a_3 / (a_3 + b_3)]    

    part b)

    We are asked to find the probability of white ball given that it pulled from bag B. So if we express it in notation we are asked for P ( B / W). We will use conditional probability to answer our question:

                               P ( B / W ) = P ( W & B ) / P (W)

                               P ( W & B ) = a_2 / (a_2 + b_2)

    Hence,

    P ( A / W ) = [a_2 / (a_2 + b_2)] / [a_1 / (a_1 + b_1) + a_2 / (a_2 + b_2) + a_3 / (a_3 + b_3)]    

    part c)

    We are asked to find the probability of white ball given that it pulled from bag C. So if we express it in notation we are asked for P ( C / W). We will use conditional probability to answer our question:

                               P ( C / W ) = P ( W & C ) / P (W)

                               P ( W & C ) = a_3 / (a_3 + b_3)

    Hence,

    P ( A / W ) = [a_3 / (a_3 + b_3)] / [a_1 / (a_1 + b_1) + a_2 / (a_2 + b_2) + a_3 / (a_3 + b_3)]    

         

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