Question

a manufacturing machine has a 7% defect rate.
if 9 items are chosen at random, what is the probability that at least one will have a defect? give your answer as a fraction or decimal out to at least 4 places.

Explanation:

Step1: Calculate the probability of no defect

The probability of an item having no defect is \(p = 1 - 0.07=0.93\).
For \(n = 9\) independent items, the probability that all 9 have no defect is \(P(X = 0)=0.93^{9}\).
Using a calculator, \(0.93^{9}\approx0.5204\).

Step2: Calculate the probability of at least one defect

The probability of at least one defect is \(P(X\geq1)=1 - P(X = 0)\).
Substitute \(P(X = 0)\approx0.5204\) into the formula: \(P(X\geq1)=1 - 0.5204 = 0.4796\).

Answer:

\(0.4796\)