---
title: "Dependent events; defective bolt from machine B"
subject: "Probability and Statistics"
chapter: 1
key: "semester_5/probability_and_statistics:bolt-factory-defective-machine-b"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/bolt-factory-defective-machine-b
site: BCT Engineering
---

# Dependent events; defective bolt from machine B

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

- Asked in: 2068 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/bolt-factory-defective-machine-b
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-1

## Question

Define dependent and independent events with examples. In a bolt factory, machines A, B and C manufacture 25%, 35% and 40% of the total respectively. Of their output 5, 4 and 2 percent are defective bolts. A bolt is drawn at random from the product and is found to be defective. What is the probability that it was manufactured from the machine B?

## Short answer

Independent: occurrence doesn't change probability; dependent: occurrence changes probability. P(B|defective) = (0.35×0.04) / (0.25×0.05 + 0.35×0.04 + 0.40×0.02) = 0.406.

## Answer

#### Independent events
Two events are independent if the occurrence of one does not change the probability of the other: P(A ∩ B) = P(A) P(B). Example: tossing two different coins.

#### Dependent events
Two events are dependent if the occurrence of one changes the probability of the other: P(A ∩ B) = P(A) P(B|A). Example: drawing two cards one after the other without replacement.

#### Bolt problem
Let A, B, C be the events that the bolt came from machine A, B, C, and D be the event that the bolt is defective.

| Machine | P(machine) | P(D given machine) | Product |
|---|---|---|---|
| A | 0.25 | 0.05 | 0.0125 |
| B | 0.35 | 0.04 | 0.0140 |
| C | 0.40 | 0.02 | 0.0080 |
| Total | | | 0.0345 |

```text
P(D) = 0.0125 + 0.0140 + 0.0080 = 0.0345
P(B|D) = P(B) P(D|B) / P(D) = 0.0140 / 0.0345 = 0.4058
```

**Answer:** the probability that the defective bolt was made by machine B is about 0.406 (28/69).

## Check yourself

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

1. Which is an example of independent events?
   A. Tossing two different coins
   B. Taking two exam papers
   C. Drawing two cards without replacement
   D. Heads and tails in one toss
   Answer: A. Outcome of one coin does not affect the other. (Key point: Independent events)

2. Dependent events satisfy which formula?
   A. P(A and B) = 0
   B. P(A and B) = P(A)P(B|A)
   C. P(A or B) = P(A)P(B)
   D. P(A and B) = P(A)P(B) always
   Answer: B. B depends on A, so conditional probability is used. (Key point: Dependent events)

3. What is P(defective)?
   A. 0.04
   B. 0.0125
   C. 0.0140
   D. 0.0345
   Answer: D. 0.0125 + 0.0140 + 0.0080 = 0.0345. (Key point: Total defect)

4. What is P(machine B | defective)?
   A. 0.04
   B. 0.35
   C. 0.4058
   D. 0.0140
   Answer: C. 0.0140/0.0345 = 0.406. (Key point: Posterior)

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