Probability in Quality Control

00:00 6 marks

A factory produces electronic components. It is known that \(5\%\) of components produced by machine A are defective, \(3\%\) from machine B are defective, and \(4\%\) from machine C are defective. Machine A produces \(40\%\) of the components, machine B produces \(35\%\), and machine C produces \(25\%\). A component is selected at random.

(a) Find the probability that the selected component is defective. [3]

(b) Given that the selected component is defective, find the probability that it was produced by machine A. [3]

Part a

5 minutes 3 marks

Question

Find the probability that the selected component is defective. [3]

Hint Hint

Hint

Multiply each machine’s defect rate by its production proportion (as decimals) and sum the results.

Answer Answer
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Explanation Explanation

Answer Answer

Complete Answer

\(P(D)=0.05\times0.40+0.03\times0.35+0.04\times0.25=0.0405\)

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Part b

5 minutes 3 marks

Question

Given that the selected component is defective, find the probability that it was produced by machine A. [3]

Hint Hint

Hint

Use Bayes’ theorem: \(P(A|D)=\frac{P(D|A)P(A)}{P(D)}\).

Answer Answer
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Explanation Explanation

Answer Answer

Complete Answer

\(P(A|D)=\frac{0.05\times0.40}{0.0405}=\frac{40}{81}\) (approximately 0.494)

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