Bayes' theorem — A/L Combined Mathematics

How do I practise Bayes' theorem?

1 question with a worked solution

Check before you answer: Partition Identification: Stating the mutually exclusive and exhaustive prior events B₁, B₂, ..., B_k explicitly.

1. Two factories X and Y produce identical light bulbs. Factory X produces 60% of the total output with a 2% defect rate, while Factory Y produces 40% with a 5% defect rate. If a randomly inspected bulb is found to be defective, what is the probability that it was manufactured by Factory X?
   1. A) 3/8 (0.375)
   2. B) 1/2 (0.500)
   3. C) 3/5 (0.600)
   4. D) 5/8 (0.625)
   Check my answer

   Correct answer: A) 3/8 (0.375)

   Let D be defective. Prior probabilities: P(X) = 0.60, P(Y) = 0.40. Conditional probabilities: P(D|X) = 0.02, P(D|Y) = 0.05. Total probability of defect: P(D) = P(X)P(D|X) + P(Y)P(D|Y) = (0.60 × 0.02) + (0.40 × 0.05) = 0.012 + 0.020 = 0.032. By Bayes' Theorem: P(X|D) = P(X)P(D|X) / P(D) = 0.012 / 0.032 = 12 / 32 = 3/8 = 0.375.

   Show worked solution

   Step 1: P(D) = (0.60 × 0.02) + (0.40 × 0.05) = 0.012 + 0.020 = 0.032. Step 2: P(X|D) = 0.012 / 0.032 = 3 / 8 = 0.375. Answer: 3/8 (0.375).
   Modelled on: the A/L Combined Mathematics examination pattern

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Where does Bayes' theorem fit in the syllabus?

Year 2 · Term 3