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Bayes' theorem; DVD players warranty

Short answer

P(brand 1 and repair) = 0.50 × 0.25 = 0.125. Total probability of repair = 0.50×0.25 + 0.30×0.20 + 0.20×0.10 = 0.205.

Explain Baye's theorem. A chain of video stores sells three different brands of DVD players; Of its DVD players sales, 50% are brand 1 (the least expensive), 30% are brand 2, and 20% are brand 3. Each manufacturer offers a 1-year warranty on parts and labor. It is known that 25% of brand 1's DVD players require warranty repair work, where the corresponding percentages for brands 2 and 3 are 20% and 10%, respectively. a) What is the probability that a randomly selected purchaser has bought a brand 1 DVD players that will need repair while under warranty? b) What is the probability that a randomly selected purchaser has a DVD player that will need repair while under warranty?

Bayes' theorem

Let E₁, E₂, …, Eₙ be mutually exclusive and exhaustive events with P(Eᵢ) > 0, and let A be any event that can occur only with one of them. Then

P(Ei | A) = P(Ei) P(A | Ei) / Σ P(Ej) P(A | Ej)

It revises a prior probability P(Eᵢ) into a posterior probability after the event A is known to have occurred. The denominator is the total probability of A.

Problem

Let B₁, B₂, B₃ be the brands and R the event that the player needs warranty repair.

P(B1) = 0.50   P(R|B1) = 0.25
P(B2) = 0.30   P(R|B2) = 0.20
P(B3) = 0.20   P(R|B3) = 0.10

a) Brand 1 and needs repair

P(B1 ∩ R) = P(B1) P(R|B1) = 0.50 × 0.25 = 0.125

b) Needs repair (total probability)

P(R) = (0.50)(0.25) + (0.30)(0.20) + (0.20)(0.10)
     = 0.125 + 0.060 + 0.020
     = 0.205

Answer: (a) 0.125, (b) 0.205.

✓ Answer checked by Claude Sonnet 5.5

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