Question

In a bulb factory, machines A, B and C manufacture 60%, 30% and 10% bulbs respectively. 1%, 2% and 3% of the bulbs produced respectively by A, B and C are found to be defective. A bulb is picked up at random from the total production and found to be defective. Find the probability that this bulb was produced by the machine A.

Solution

Let E₁, E₂ , E₃ be the events of the bulb is produced by machines A, B, C respectively.

therefore space space space straight P left parenthesis straight E subscript 1 right parenthesis space equals space 60 over 100 space equals space 6 over 10 comma space space space straight P left parenthesis straight E subscript 2 right parenthesis space equals space 30 over 100 space equals space 3 over 10 comma space space straight P left parenthesis straight E subscript 3 right parenthesis space equals space 10 over 100 space equals 1 over 10

Let E be the event of bulb being defective.

therefore space space space space space straight P left parenthesis straight E space vertical line space straight E subscript 1 right parenthesis space equals space 1 over 100 comma space space space straight P left parenthesis straight E space vertical line space straight E subscript 2 right parenthesis space equals space 2 over 100 comma space space straight P left parenthesis straight E space vertical line thin space straight E subscript 3 right parenthesis space equals space 3 over 100

Required probability =

straight P left parenthesis straight E subscript 1 space vertical line thin space straight E right parenthesis

equals space fraction numerator straight P left parenthesis straight E subscript 1 right parenthesis. space space straight P left parenthesis straight E vertical line straight E subscript 1 right parenthesis over denominator straight P left parenthesis straight E subscript 1 right parenthesis. space straight P left parenthesis straight E space vertical line thin space 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 E space vertical line thin space 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 E space vertical line thin space straight E subscript 3 right parenthesis end fraction equals space fraction numerator open parentheses begin display style 6 over 10 end style close parentheses space open parentheses begin display style 1 over 100 end style close parentheses over denominator open parentheses begin display style 6 over 10 end style close parentheses space open parentheses begin display style 1 over 100 end style close parentheses space plus space open parentheses begin display style 3 over 10 end style close parentheses open parentheses begin display style 2 over 100 end style close parentheses plus open parentheses begin display style 1 over 10 end style close parentheses space open parentheses begin display style 3 over 100 end style close parentheses space end fraction equals space fraction numerator 6 over denominator 6 plus 6 plus 3 end fraction equals space 6 over 15 space equals 2 over 5

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