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pmf conditions; binomial defective items

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.

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.

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

✓ Answer checked by Claude Sonnet 5.5

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