---
title: "Bayes' theorem; defective thermostat supplier"
subject: "Probability and Statistics"
chapter: 1
key: "semester_5/probability_and_statistics:bayes-thermostat-company-a"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/bayes-thermostat-company-a
site: BCT Engineering
---

# Bayes' theorem; defective thermostat supplier

Probability and Statistics · Chapter 1: Descriptive Statistics and Basic Probability

- Asked in: 2070 Chaitra, 2+4 marks
- Video: [Bayes Theorem Probability of Defective Bolt from three machines  IIT JEE CBSE 12 NCERT](https://www.youtube.com/watch?v=B8KcbYf_IMQ) — Anil Kumar
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/bayes-thermostat-company-a
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-1

## Question

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.

## 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.

## Answer

#### Bayes' theorem
If E₁, E₂, …, Eₙ are mutually exclusive and exhaustive events and A is any event with P(A) > 0, then

```text
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:

```text
P(D) = 0.0035 + 0.0020 + 0.0015 = 0.0070
```

Probability that company A supplied it:

```text
P(A|D) = 0.0035 / 0.0070 = 0.5
```

**Answer:** P(A | defective) = 0.5.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What does Bayes' theorem give?
   A. The sum of all prior probabilities
   B. The chance of two independent events together
   C. The probability of the effect only
   D. The probability of a cause after the effect is observed
   Answer: D. It revises the probability of a cause Ei after event A has been observed. (Key point: Bayes statement)

2. What share of thermostats does company C supply?
   A. 5%
   B. 30%
   C. 10%
   D. 20%
   Answer: C. 100% - 70% - 20% = 10%. (Key point: Missing share)

3. What is P(defective)?
   A. 0.0200
   B. 0.0070
   C. 0.0035
   D. 0.015
   Answer: B. 0.0035 + 0.0020 + 0.0015 = 0.0070. (Key point: Total defect probability)

4. What is P(A | defective)?
   A. 0.005
   B. 0.25
   C. 0.70
   D. 0.5
   Answer: D. 0.0035/0.0070 = 0.5. (Key point: Posterior)

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