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Q. 11   A manufacturer has three machine operators A,B and C. The first operator A produces 1^{o}/_{o} defective items, where as the other two operators B and C produce 5^{o}/_{o} and 7^{o}/_{o} defective items respectively. A is on the job for 50^{o}/_{o} of the time, B is on the job for 30^{o}/_{o} of the time and C is on the job for 20^{o}/_{o}  of the time. A defective item is produced, what is the probability that it was produced by A?

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Let A: time consumed by machine A =50\%

       B: time consumed by machine B=30\%

       C: time consumed by machine C =20\%

       Total drivers = 12000

P(A)=\frac{50}{100}=\frac{1}{2}

P(B)=\frac{30}{100}=\frac{3}{10}

P(C)=\frac{20}{100}=\frac{1}{5}

  D: Event of producing defective items 

  P(D|A)= \frac{1}{100}

P(D|B)= \frac{5}{100}

P(D|C)= \frac{7}{100}

P(A|D)=\frac{P(A).P(D|A)}{P(B).P(D|B)+P(A).P(D|A)+P(C).P(D|C)}

 P(A|D)=\frac{\frac{1}{2}\times \frac{1}{100}}{\frac{1}{2}\times \frac{1}{100}+\frac{3}{10}\times \frac{5}{100}+\frac{1}{5}\times \frac{7}{100}}

P(A|D)=\frac{\frac{1}{2}\times \frac{1}{100}}{\frac{1}{100} (\frac{1}{2}+\frac{3}{2}+\frac{7}{5})}

P(A|D)=\frac{\frac{1}{2}}{ (\frac{17}{5})}

P(A|D)= \frac{5}{34}

Hence, the probability that defective item was produced by A = 

                                                                                                  P(A|D)= \frac{5}{34}

Posted by

seema garhwal

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