Questions: A manufacturing machine has a 5% defect rate. If 8 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 8 items are chosen at random from a manufacturing machine with a defect rate of \(5\%\).

To find the probability of at least one defect, we first calculate the probability of having zero defects. The probability of exactly \(x\) successes (defects) in \(n\) trials can be expressed using the binomial distribution formula:

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

Where:

• \(n = 8\) (number of trials)
• \(x = 0\) (number of successes)
• \(p = 0.05\) (probability of success)
• \(q = 1 - p = 0.95\) (probability of failure)

Substituting the values, we find:

\[ P(X = 0) = \binom{8}{0} \cdot (0.05)^0 \cdot (0.95)^8 = 1 \cdot 1 \cdot 0.6634 = 0.6634 \]

Thus, the probability of zero defects is:

\[ P(\text{zero defects}) = 0.6634 \]

Using the complement rule, the probability of at least one defect is given by:

\[ P(\text{at least one defect}) = 1 - P(\text{zero defects}) \]

Substituting the previously calculated value:

\[ P(\text{at least one defect}) = 1 - 0.6634 = 0.3366 \]

Final Answer