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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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