Paper 4 · conditional probability / Bayes theorem

A manufacturing plant uses three machines, A, B and C, contributing 40%, 45% and 15% of daily output respectively, with defect rates of 4%, 3% and 1% for A, B and C respectively. Given that a randomly chosen tin is defective, what is the probability, to 3 decimal places, that it came from machine B?

Model answer

0.435

Also accepted: 0.436, 27/62

Explanation: By Bayes' theorem,

P(B\mid\text{defective}) = \dfrac{P(B)P(\text{defective}\mid B)}{P(\text{defective})} = \dfrac{0.45\times0.03}{0.031} = \dfrac{0.0135}{0.031} = \dfrac{27}{62} \approx 0.435

Derived from ZIMSEC Maths Paper 1, June 2007, Q1 (Statistics)

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