---
title: "Hypergeometric: 4 batteries from 24"
subject: "Probability and Statistics"
chapter: 2
key: "semester_5/probability_and_statistics:batteries-4-of-24-defect"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/batteries-4-of-24-defect
site: BCT Engineering
---

# Hypergeometric: 4 batteries from 24

Probability and Statistics · Chapter 2: Discrete Probability Distribution

- Asked in: 2074 Asoj, 5 marks; 2071 Chaitra, 5 marks (asked 2 times)
- Video: [An Introduction to the Hypergeometric Distribution](https://www.youtube.com/watch?v=L2KMttDm3aY) — jbstatistics
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/batteries-4-of-24-defect
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-2

## Question

A quality control engineer inspects a random sample of 4 batteries from each lot of 24 car batteries that is ready to shipment. If such a lot contain six batteries with slight defects. What are the probabilities that the inspector's sample will contain:
i) None of the batteries with defect?
ii) At least two of the batteries with defects?
iii) At most three of the batteries with defect?

## Short answer

Use hypergeometric distribution: (i) 0.2880, (ii) 0.2513, (iii) 0.9986.

## Answer

N = 24, K = 6 defective, 18 good, sample n = 4 without replacement, so the number of defectives X is **hypergeometric**:

P(X = x) = C(6, x) · C(18, 4 − x) / C(24, 4), with C(24, 4) = 10626.

```text
x   C(6,x)·C(18,4-x)      P(X = x)
0   1 × 3060  = 3060      0.2880
1   6 × 816   = 4896      0.4608
2   15 × 153  = 2295      0.2160
3   20 × 18   = 360       0.0339
4   15 × 1    = 15        0.0014
```

#### i) None defective
P(X = 0) = 3060 / 10626 = **0.2880**

#### ii) At least two defective
P(X ≥ 2) = 1 − P(0) − P(1) = 1 − 0.2880 − 0.4608 = **0.2513**
(Direct: (2295 + 360 + 15) / 10626 = 2670 / 10626 = 0.2513.)

#### iii) At most three defective
P(X ≤ 3) = 1 − P(X = 4) = 1 − 15/10626 = 1 − 0.0014 = **0.9986**

**Answer: (i) 0.2880, (ii) 0.2513, (iii) 0.9986**

## Check yourself

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

1. Which formula applies?
   A. Hypergeometric C(6,x)C(18,4-x)/C(24,4)
   B. Poisson
   C. Binomial with p=6/24 always
   D. Geometric
   Answer: A. Without replacement, finite lot. (Key point: Distribution)

2. P(X=0)?
   A. 0.0014
   B. 0.2160
   C. 0.2880
   D. 0.4608
   Answer: C. 3060/10626 = 0.2880. (Key point: None defective)

3. What is P(X>=2)?
   A. 0.2513
   B. 0.7487
   C. 0.0339
   D. 0.2160
   Answer: A. 1 - 0.2880 - 0.4608 = 0.2513. (Key point: At least two)

4. How is P(X<=3) found most simply?
   A. P(0)+P(1) only
   B. P(3)
   C. 1 - P(0)
   D. 1 - P(X=4)
   Answer: D. Only X=4 is excluded: 1 - 15/10626 = 0.9986. (Key point: At most three)

## Close variants set in other exams

- [Hypergeometric: 3 batteries from 24](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Abatteries-3-of-24-defect) — Probability and Statistics, chapter 2, asked 4×