Question

A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black balls. One of the two bags is selected at random and a ball is drawn from the bag which is found to be red. Find the probability that the ball is drawn from the first bag.

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Solution

Let E₁ be the event of choosing the first bag, E₂ the event of choosing the second bag and A be the event of drawing a red ball.

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

Now,

straight P left parenthesis straight A space vertical line space straight E subscript 1 right parenthesis space equals space straight P left parenthesis drawing space straight a space red space ball space from space first space bag right parenthesis space equals 4 over 8 space equals 1 half

and

straight P left parenthesis straight A space vertical line space straight E subscript 2 right parenthesis space equals space straight P left parenthesis drawing space straight a space red space ball space from space second space bag right parenthesis space equals space 2 over 8 space equals space 1 fourth

therefore space space space straight P left parenthesis straight E subscript 1 space vertical line space straight A right parenthesis space equals space fraction numerator straight P left parenthesis straight E subscript 1 right parenthesis space straight P left parenthesis straight A vertical line straight E subscript 1 right parenthesis over denominator straight P left parenthesis straight E subscript 1 right parenthesis thin space straight P left parenthesis straight A space vertical line thin space straight E subscript 1 right parenthesis space plus space straight P left parenthesis straight E subscript 2 right parenthesis thin space straight P left parenthesis straight A space vertical line thin space straight E subscript 2 right parenthesis end fraction

equals space fraction numerator begin display style 1 half end style cross times begin display style 1 half end style over denominator begin display style 1 half end style cross times begin display style 1 half end style plus begin display style 1 half end style cross times begin display style 1 fourth end style end fraction space equals fraction numerator begin display style 1 fourth end style over denominator begin display style 1 fourth end style plus begin display style 1 over 8 end style end fraction space equals space fraction numerator begin display style 1 fourth end style over denominator begin display style fraction numerator 2 plus 1 over denominator 8 end fraction end style end fraction space equals fraction numerator begin display style 1 fourth end style over denominator begin display style 3 over 8 end style end fraction space equals space 1 fourth cross times 8 over 3 space equals 2 over 3.

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