--- title: "Poisson: generators failing in warranty" subject: "Probability and Statistics" chapter: 2 key: "semester_5/probability_and_statistics:generators-warranty-poisson" source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/generators-warranty-poisson site: BCT Engineering --- # Poisson: generators failing in warranty Probability and Statistics · Chapter 2: Discrete Probability Distribution - Asked in: 2071 Shrawan, 5 marks - Video: [Poisson Variate - Problem](https://www.youtube.com/watch?v=xwU8x1-Z3_k) — Shailesh Shetty - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/generators-warranty-poisson - Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-2 ## Question A heavy machinery manufacturer has 3840 large generators in the field that are under warranty. If the probability is 1/1200 that any one will fail during the given year, find the probability: i) That exactly 3 generators will fail during the given year? ii) That between 2 and 6 are fail during the given year? ## Short answer Poisson approximation with λ=np=3.2. P(exactly 3 fail)=0.2226; P(between 2 and 6 inclusive)=0.7842. ## Answer n = 3840 generators, p = 1/1200 (small), so the number of failures X is approximately **Poisson** with λ = np = 3840 / 1200 = 3.2, P(X = x) = e⁻³·² (3.2)ˣ / x!, e⁻³·² = 0.040762. ```text x P(X = x) 2 0.040762 × 3.2²/2 = 0.208702 3 0.040762 × 3.2³/6 = 0.222616 4 0.222616 × 3.2/4 = 0.178093 5 0.178093 × 3.2/5 = 0.113979 6 0.113979 × 3.2/6 = 0.060789 ``` #### i) Exactly 3 fail P(X = 3) = **0.2226** #### ii) Between 2 and 6 fail Taking the limits as included (2 ≤ X ≤ 6): P = 0.2087 + 0.2226 + 0.1781 + 0.1140 + 0.0608 = **0.7842** (If "between" is read as strictly between, 3 ≤ X ≤ 5, the answer is 0.2226 + 0.1781 + 0.1140 = 0.5147.) **Answer: (i) 0.2226, (ii) 0.7842** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Poisson is used because: A. p is large B. n is small C. X is continuous D. n = 3840 is large and p = 1/1200 is small Answer: D. Large n, small p. (Key point: Why Poisson) 2. lambda is: A. 1/1200 B. 3840/1200 = 3.2 C. 3840 D. 3.2^2 Answer: B. lambda = np. (Key point: Lambda) 3. P(X = 3) is: A. e^-3.2 (3.2)^3 = 1.34 B. e^-3.2 (3.2)^3 / 3! = 0.2226 C. e^-3 (3.2)^3/3! D. 3.2^3/3! Answer: B. Apply the Poisson formula with lambda = 3.2. (Key point: Exactly 3) 4. Taking 'between 2 and 6' as inclusive, the probability is: A. P(3)+P(4)+P(5) = 0.5147 always B. P(6) = 0.0608 C. 1 - P(2) = 0.79 D. P(2)+P(3)+P(4)+P(5)+P(6) = 0.7842 Answer: D. Include both limits 2 and 6 as the answer does; the strict reading gives 0.5147. (Key point: Between 2 and 6) ## Related in Probability and Statistics - [Bayes' theorem; DVD players warranty](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Abayes-dvd-warranty) — Probability and Statistics, chapter 1 - [Normal: machinery assembly time](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Aassembly-time-normal-12-9) — Probability and Statistics, chapter 3 - [Poisson process; intersection accidents](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Apoisson-process-accidents-intersection) — Probability and Statistics, chapter 2 - [Poisson: error pages in 400-page novel](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Apublisher-typo-pages-400) — Probability and Statistics, chapter 2 - [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× - [Poisson: bad reaction among 2000 (a, b)](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Apoisson-injection-reaction-2000) — Probability and Statistics, chapter 2, asked 2× ## Related in other subjects - [Prove mean annual flood has 2.33-year return period](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Agumbel-mean-flood-2-33yr) — Engineering Hydrology, chapter 6 - [Shunt motor speed with 50 Ω field resistance (1200 rpm)](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines_i%3Ashunt-motor-250v-25a-half-load) — Electrical Machines I, chapter 4, asked 5× - [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