---
title: "pmf conditions; binomial defective items"
subject: "Probability and Statistics"
chapter: 2
key: "semester_5/probability_and_statistics:pmf-conditions-defective-8-items"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/pmf-conditions-defective-8-items
site: BCT Engineering
---

# pmf conditions; binomial defective items

Probability and Statistics · Chapter 2: Discrete Probability Distribution

- Asked in: 2068 Chaitra, 2+2+2 marks
- Video: [Binomial distribution probability (solve with easy steps)](https://www.youtube.com/watch?v=vqojqDds_eo) — Shrenik Jain
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/pmf-conditions-defective-8-items
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-2

## Question

Write any two conditions that a function is a probability mass function. It is found that 10% of the items produced by a company are defective. Out of 8 items chosen, using binomial distribution, find the probability of: (i) no defective item (ii) at least one defective item.

## Short answer

A valid pmf must satisfy p(x)>=0 for all x and sum p(x)=1. For B(8,0.1), probability of no defective is 0.9^8=0.4305; at least one defective is 1-0.4305=0.5695.

## Answer

#### Conditions for a probability mass function
A function p(x) of a discrete random variable X is a pmf if
1. p(x) ≥ 0 for every value x, and
2. Σ p(x) = 1 (the sum over all possible x).

#### Binomial problem
Each item is defective with p = 0.10 (q = 0.90). For n = 8 items, X = number of defectives ~ B(8, 0.1):

P(X = x) = C(8, x)(0.1)ˣ(0.9)⁸⁻ˣ

**(i) No defective item**
P(X = 0) = (0.9)⁸ = **0.4305**

**(ii) At least one defective item**
P(X ≥ 1) = 1 − P(X = 0) = 1 − 0.4305 = **0.5695**

**Answer: (i) 0.4305, (ii) 0.5695**

## Check yourself

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

1. A valid pmf requires:
   A. p(x) <= 0 and sum = 1
   B. p(x) >= 0 and sum of p(x) = 1
   C. Sum of x p(x) = 1
   D. p(x) > 1 for some x
   Answer: B. Probabilities are non-negative and total 1. (Key point: pmf conditions)

2. P(none of 8 defective) with p = 0.1 is:
   A. 0.1^8
   B. 8 x 0.1
   C. 1 - 0.9^8
   D. 0.9^8 = 0.4305
   Answer: D. All eight must be good. (Key point: No defective)

3. P(at least one defective) is:
   A. 1 - 0.4305 = 0.5695
   B. 0.4305
   C. 8 x 0.1 = 0.8
   D. 0.1^8
   Answer: A. Complement of none. (Key point: At least one)

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