---
title: "Binomial: defective lenses in packet"
subject: "Probability and Statistics"
chapter: 2
key: "semester_5/probability_and_statistics:lenses-defective-packet-of-10"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/lenses-defective-packet-of-10
site: BCT Engineering
---

# Binomial: defective lenses in packet

Probability and Statistics · Chapter 2: Discrete Probability Distribution

- Asked in: 2072 Chaitra, 5 marks
- Video: [Poisson Distribution | Defective Blades Problem | BCS301 Module 1 Q1b Model QP-I](https://www.youtube.com/watch?v=Y_M7xbVwNM8) — Mathematics Tutor
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/lenses-defective-packet-of-10
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-2

## Question

In certain factory turning out optical lenses, there is a small chance, 1/500 for any lens to be defective. The lenses are supplied in packets of 10 each. What is the probability that a packet will contain
i) No defective lens
ii) At least one defective lenses
iii) At most two defective lenses

## Short answer

Binomial n=10, p=0.002: P(no defective)=0.9802; P(at least one)=0.0198; P(at most two)≈1.0000.

## Answer

Each lens is defective with probability p = 1/500 = 0.002, independently. For a packet of n = 10, the number of defective lenses X ~ B(10, 0.002):

P(X = x) = C(10, x)(0.002)ˣ(0.998)¹⁰⁻ˣ

```text
P(0) = 0.998¹⁰               = 0.98018
P(1) = 10 × 0.002 × 0.998⁹   = 0.01964
P(2) = 45 × 0.002² × 0.998⁸  = 0.00018
```

#### i) No defective lens
P(X = 0) = **0.9802**

#### ii) At least one defective lens
P(X ≥ 1) = 1 − P(0) = 1 − 0.98018 = **0.0198**

#### iii) At most two defective lenses
P(X ≤ 2) = 0.98018 + 0.01964 + 0.00018 = **0.99999 ≈ 1.0000**

(Poisson approximation with λ = np = 0.02 gives almost the same values: 0.9802, 0.0198, 1.0000.)

**Answer: (i) 0.9802, (ii) 0.0198, (iii) ≈ 1.0000**

## Check yourself

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

1. The number of defective lenses in a packet follows:
   A. B(500, 0.1)
   B. B(10, 0.002)
   C. Poisson with lambda 10
   D. B(10, 0.998)
   Answer: B. n = 10 lenses with defect probability 1/500 = 0.002. (Key point: Model)

2. P(none defective) is:
   A. 10 x 0.002 = 0.02
   B. 0.998^10 = 0.9802
   C. 0.002^10
   D. 1 - 0.998^10
   Answer: B. All 10 must be good: q^n. (Key point: No defective)

3. P(at least one defective) is found by:
   A. P(1) only
   B. P(0) = 0.9802
   C. 1 - P(2)
   D. 1 - P(0) = 0.0198
   Answer: D. Use the complement of none. (Key point: At least one)

4. P(X <= 2) is approximately:
   A. 1.0000
   B. 0.0198
   C. 0.5
   D. 0.9802
   Answer: A. P(0)+P(1)+P(2) = 0.99999, nearly certain. (Key point: At most two)

## Related in Probability and Statistics

- [pmf conditions; binomial defective items](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Apmf-conditions-defective-8-items) — Probability and Statistics, chapter 2
- [Dependent events; defective bolt from machine B](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Abolt-factory-defective-machine-b) — Probability and Statistics, chapter 1
- [Binomial: thick coating on 15 chips](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Aic-chips-thick-coating-15) — Probability and Statistics, chapter 2
- [Bayes' theorem; defective thermostat supplier](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Abayes-thermostat-company-a) — Probability and Statistics, chapter 1
- [Normal: percentage of defective washers](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Awashers-defective-percentage-normal) — Probability and Statistics, chapter 3
- [Conditional probability; voltage regulator suppliers](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Aconditional-probability-voltage-regulators) — Probability and Statistics, chapter 1, asked 2×

## Related in other subjects

- [Laplace: current supplied by source, 10 V](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Alaplace-source-current-10v-1ohm-1h) — Electrical Circuit Theory, chapter 4
- [Types of USB packets](https://www.bctengineer.com/question.md?key=semester_5%2Finstrumentation_ii%3Ausb-packet-types) — Instrumentation II, chapter 3, asked 3×
- [Lossless one-port properties; Foster II and Cauer II](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Alossless-one-port-foster-cauer-ii) — Filter Design, chapter 4
- [Lossless one-port properties; Cauer I and Foster II](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Alossless-one-port-cauer-foster-ii) — Filter Design, chapter 4