- 2070 Chaitra · 2+4 marks
Bayes' theorem; defective thermostat supplier
Short answer
Bayes' theorem: P(Ei|A) = P(Ei)P(A|Ei) / Σ P(Ej)P(A|Ej). Here total defect probability P(D)=0.007, so P(A supplied | defective) = 0.0035 / 0.007 = 0.5.
State Baye's theorem. A manufacturer of air-conditioning units purchases 70% of its thermostats from company A, 20% from company B and the rest from company C. Past experience shows that 0.5% of company A's thermostats, 1% of company B's thermostats and 1.5% of company C's thermostats are likely to be defective. An air-conditioning unit randomly selected from this manufacturer's production line was found to have a defective thermostat. Find the probability that company A supplied the defective thermostat.
Bayes' theorem
If E₁, E₂, …, Eₙ are mutually exclusive and exhaustive events and A is any event with P(A) > 0, then
P(Ei | A) = P(Ei) P(A | Ei) / Σ P(Ej) P(A | Ej)
It gives the probability of a cause Eᵢ after the effect A has been observed.
Thermostat problem
Company C supplies the remaining 100% − 70% − 20% = 10%. Let D be the event that a thermostat is defective.
| Company | P(company) | P(D given company) | Product |
|---|---|---|---|
| A | 0.70 | 0.005 | 0.0035 |
| B | 0.20 | 0.010 | 0.0020 |
| C | 0.10 | 0.015 | 0.0015 |
| Total | 0.0070 |
Total probability of a defect:
P(D) = 0.0035 + 0.0020 + 0.0015 = 0.0070
Probability that company A supplied it:
P(A|D) = 0.0035 / 0.0070 = 0.5
Answer: P(A | defective) = 0.5.
✓ Answer checked by Claude Sonnet 5.5
Related questions
- Bayes' theorem; DVD players warranty · Probability and Statistics, chapter 1
- Mutually exclusive vs equally likely; Bayes topper · Probability and Statistics, chapter 1
- Event types; airport weapon detection Bayes · Probability and Statistics, chapter 1
- pmf conditions; binomial defective items · Probability and Statistics, chapter 2
- Short note: Air conditioning · Building Technology, chapter 1 · asked 3×
- Ventilation versus air-conditioning · Building Technology, chapter 1
- Building types and essentials of air conditioning · Building Technology, chapter 1
- Air-conditioning: definition and scope · Applied Thermodynamics, chapter 4