---
title: "Poisson: bad reaction among 2000 (a, b)"
subject: "Probability and Statistics"
chapter: 2
key: "semester_5/probability_and_statistics:poisson-injection-reaction-2000"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/poisson-injection-reaction-2000
site: BCT Engineering
---

# Poisson: bad reaction among 2000 (a, b)

Probability and Statistics · Chapter 2: Discrete Probability Distribution

- Asked in: 2076 Chaitra, 5 marks
- Video: [BCS301 Module 1 | Poisson Distribution |Find Probability of Bad Reaction |New Model Question Paper-I](https://www.youtube.com/watch?v=RbMbXdhJ_KE) — Mathematics Tutor
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/poisson-injection-reaction-2000
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-2

## Question

If the probability that an individual suffers a bad reaction from a certain injection is 0.001, determine the probability that out of 2000 individuals
a) Exactly 3, individuals will suffer bad reaction
b) More than 2, individuals will suffer bad reaction

## Short answer

Use Poisson with λ=2: (a) exactly 3 is 0.1804, (b) more than 2 is 0.3233.

## Answer

n = 2000 is large and p = 0.001 is small, so use the **Poisson approximation** to the binomial with

λ = np = 2000 × 0.001 = 2, P(X = x) = e^(−λ) λˣ / x! = e⁻² 2ˣ / x!, with e⁻² = 0.135335.

#### a) Exactly 3 suffer a bad reaction
P(X = 3) = e⁻² · 2³ / 3! = 0.135335 × 8 / 6 = **0.1804**

#### b) More than 2 suffer a bad reaction
P(X > 2) = 1 − [P(0) + P(1) + P(2)]

```text
P(0) = e⁻²            = 0.1353
P(1) = e⁻² · 2        = 0.2707
P(2) = e⁻² · 2²/2     = 0.2707
Sum                   = 0.6767
```

P(X > 2) = 1 − 0.6767 = **0.3233**

**Answer: (a) 0.1804, (b) 0.3233**

## Check yourself

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

1. Why use Poisson here?
   A. n small
   B. p is 0.5
   C. n large, p small, so binomial is approximated with lambda = np
   D. Trials dependent
   Answer: C. Rare events in many trials. (Key point: Poisson use)

2. What is lambda?
   A. 2
   B. 2000
   C. 0.5
   D. 0.001
   Answer: A. 2000 x 0.001 = 2. (Key point: Lambda)

3. What is P(X=3)?
   A. 0.3233
   B. 0.1804
   C. 0.0902
   D. 0.2707
   Answer: B. e^-2 x 8/6 = 0.1804. (Key point: P(X=3))

4. How is P(X>2) found?
   A. P(0)+P(1)+P(2)
   B. 1 - [P(0)+P(1)+P(2)] = 0.3233
   C. 1 - P(3)
   D. P(3) only
   Answer: B. The sum to 2 is 0.6767, so P(X>2) = 0.3233. (Key point: More than 2)

## Close variants set in other exams

- [Poisson: injection reaction with distribution](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Apoisson-injection-with-pdf) — Probability and Statistics, chapter 2, asked 2×

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