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

Solution Steps

We are tasked with finding the probability that at least one item will have a defect when 9 items are chosen at random from a manufacturing machine with a defect rate of \(9\%\) or \(p = 0.09\).

To find the probability of at least one defect, we first calculate the probability of having no defects (0 defects) among the 9 items. This can be computed using the binomial probability formula:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

where:

• \(n = 9\) (the number of trials),
• \(x = 0\) (the number of successes, which in this case means no defects),
• \(p = 0.09\) (the probability of a defect),
• \(q = 1 - p = 0.91\) (the probability of no defect).

Substituting the values, we find:

\[ P(X = 0) = \binom{9}{0} \cdot (0.09)^0 \cdot (0.91)^9 = 1 \cdot 1 \cdot (0.91)^9 \approx 0.4279 \]

The probability of at least one defect is given by the complement of the probability of no defects:

\[ P(\text{at least one defect}) = 1 - P(X = 0) = 1 - 0.4279 \approx 0.5721 \]

Final Answer