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Question
An item is manufactured by three machines A, B and C. Out of the total number of items manufactured during a specified period, 50% are manufactured on machine A, 30% on Band 20% on C. 2% of the items produced on A and 2% of items produced on B are defective and 3% of these produced on C are defective. All the items stored at one godown. One item is drawn at random and is found to be defective. What is the probability that it was manufactured on machine A?
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Solution
\[\text{ Let E be the event of getting a defective item } . \]
\[\text{ We have } , \]
\[P\left( A \right) = 50 % = \frac{50}{100} = \frac{1}{2}, P\left( B \right) = 30 % = \frac{30}{100} = \frac{3}{10} and P\left( C \right) = 20 % = \frac{20}{100} = \frac{1}{5}, \]
\[P\left(E|A \right) = 2 % = \frac{2}{100} = \frac{1}{50}, P\left( E|B \right) = 2 % = \frac{2}{100} = \frac{1}{50} and P\left( E|C \right) = 3 % = \frac{3}{100}\]
\[\text{ Now } , \]
\[P\left( \text{ the defective item drawn was manufactured on machine } A \right) = \frac{P\left( A \right) \times P\left( E|A \right)}{P\left( A \right) \times P\left( E|A \right) + P\left( B \right) \times P\left( E|B \right) + P\left( C \right) \times P\left( E|C \right)}\]
\[ = \frac{\frac{1}{2} \times \frac{1}{50}}{\frac{1}{2} \times \frac{1}{50} + \frac{3}{10} \times \frac{1}{50} + \frac{1}{5} \times \frac{3}{100}}\]
\[ = \frac{\left( \frac{1}{100} \right)}{\left( \frac{1}{100} + \frac{3}{500} + \frac{3}{500} \right)}\]
\[ = \frac{\left( \frac{1}{100} \right)}{\left( \frac{5 + 3 + 3}{500} \right)}\]
\[ = \frac{\left( \frac{1}{100} \right)}{\left( \frac{11}{500} \right)}\]
\[ = \frac{500}{100 \times 11}\]
So, the probability that the defective item drawn was manufactured on machine A is \[\frac{5}{11}\]
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