Question

a manufacturing machine has a 9% defect rate.
if 6 items are chosen at random, what is the probability that at least one will have a defect?

Explanation:

Step1: Calculate the probability of no defect

The probability of an item having no defect is \(p = 1 - 0.09=0.91\).
Using the binomial probability formula \(P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}\), for \(n = 6\), \(k = 0\), \(C(6,0)=\frac{6!}{0!(6 - 0)!}=1\).
So \(P(X = 0)=1\times(0.91)^{6}\).
\(P(X = 0)=(0.91)^{6}\approx0.5679\).

Step2: Calculate the probability of at least one defect

The probability of at least one defect \(P(X\geq1)=1 - P(X = 0)\).
Substitute \(P(X = 0)\approx0.5679\) into the formula: \(P(X\geq1)=1 - 0.5679 = 0.4321\).

Answer:

\(0.4321\)