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A bag X contains 4 white balls and 2 black balls, while another bag Y contains 3 white balls and 3 black balls. Two balls are drawn (without replacement) at random from one of the bags and were found to be one white and one black. Find the probability that the balls were drawn from bag Y.


Given,
Bag X =4 white, 2 black

Bag Y=3 white, 3 black

Let A be the event of selecting one white and one black ball.

E1 = first bag selected

E2 = second bag selected

straight P left parenthesis straight E subscript 1 right parenthesis space equals 1 half space straight P left parenthesis straight E subscript 2 right parenthesis space equals 1 half

straight P left parenthesis straight A over straight E subscript 1 right parenthesis space equals space 4 over 6 space straight x space 2 over 5 space plus space 2 over 6 space straight x 4 over 5 space equals 16 over 30

straight P open parentheses straight A over straight E subscript 2 close parentheses space equals space space 3 over 6 space straight x 3 over 5 space plus 3 over 6 space straight x 3 over 5 space equals 18 over 30

straight P open parentheses straight E subscript 2 over straight A close parentheses space equals space fraction numerator straight P space left parenthesis straight E subscript 2 right parenthesis straight P space left parenthesis straight A divided by straight E subscript 2 right parenthesis over denominator straight P left parenthesis straight E subscript 1 right parenthesis straight P left parenthesis straight A divided by straight E subscript 1 right parenthesis plus straight P left parenthesis straight E subscript 2 right parenthesis straight P left parenthesis straight A divided by straight E subscript 2 right parenthesis end fraction

straight P open parentheses straight E subscript 2 over straight A close parentheses space equals space fraction numerator begin display style 1 half end style space straight x begin display style 18 over 30 end style over denominator begin display style 1 half end style space straight x begin display style 16 over 30 end style space plus begin display style 1 half end style space straight x begin display style 18 over 30 end style end fraction

space equals fraction numerator 18 over denominator 16 plus 18 end fraction

space space equals 18 over 34

space space equals 9 over 17

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