Question

In a tape-recorder factory, three machines A, B, C produce 50%, 30%, 20% of the total production. The percentage of the defective output of these machines arc 3%, 4% and 5%. A tape recorder is selected at random and is found to be defective. Find the probability that the tape recorder was produced by machine A.

Solution

Let E₁ , E₂, E₃ be the events that the tape recorder is produced by machines A, B, C respectively.

therefore space space space space straight P left parenthesis straight E subscript 1 right parenthesis space equals space 50 over 100 space equals 1 half comma space space space straight P left parenthesis straight E subscript 2 right parenthesis space equals space 30 over 100 equals space 3 over 10 comma space space straight P left parenthesis straight E subscript 3 right parenthesis space equals space 1 fifth therefore space space space space space space straight P left parenthesis straight E subscript 1 right parenthesis space plus space straight P left parenthesis straight E subscript 2 right parenthesis space plus space straight P left parenthesis straight E subscript 3 right parenthesis space equals space 1 half plus 3 over 10 plus 1 fifth space equals fraction numerator 5 plus 3 plus 2 over denominator 10 end fraction space equals 1

∴ events are exhaustive and mutually exclusive
Let D be the event of the tape-recorder being defective.

therefore space space space straight P left parenthesis straight D divided by straight E subscript 1 right parenthesis space equals space probability space that space it space produced space by space machine space straight A space equals space 3 over 100

straight P left parenthesis straight D divided by straight E subscript 2 right parenthesis space equals space 4 over 100 comma space space space straight P left parenthesis straight D divided by straight E subscript 3 right parenthesis space equals space 5 over 100

therefore space space space space straight P left parenthesis straight E subscript 1 divided by straight D right parenthesis space equals space fraction numerator straight P left parenthesis straight E subscript 1 right parenthesis. space space straight P left parenthesis straight D divided by straight E subscript 1 right parenthesis over denominator straight P left parenthesis straight E subscript 1 right parenthesis. space straight P left parenthesis straight D divided by straight E subscript 1 right parenthesis space plus space straight P left parenthesis straight E subscript 2 right parenthesis. space straight P left parenthesis straight D divided by straight E subscript 2 right parenthesis space plus space straight P left parenthesis straight E subscript 3 right parenthesis. space straight P left parenthesis straight D divided by straight E subscript 3 right parenthesis end fraction space space space space space space space space space space space space space space space space space space space space space space space space space

equals space fraction numerator begin display style 1 half end style cross times begin display style 3 over 100 end style over denominator begin display style 1 half end style cross times begin display style 3 over 100 end style plus begin display style 3 over 10 end style cross times begin display style 4 over 100 end style plus begin display style 1 fifth end style cross times begin display style 5 over 100 end style end fraction space equals fraction numerator begin display style 3 over 2 end style over denominator begin display style 3 over 2 end style plus begin display style 6 over 5 end style plus 1 end fraction

equals fraction numerator begin display style 3 over 2 end style over denominator begin display style fraction numerator 15 plus 12 plus 10 over denominator 10 end fraction end style end fraction space equals space 3 over 2 cross times 10 over 37 space equals space 15 over 37.

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