Questions: Suppose a shipment of 160 electronic components contains 4 defective components. To determine whether the shipment should be accepted, a quality-control engineer randomly selects 4 of the components and tests them. If 1 or more of the components is defective, the shipment is rejected. What is the probability that the shipment is rejected? The probability that the shipment is rejected is (Round to four decimal places as needed.)

Solution Steps

We are tasked with determining the probability that a shipment of 160 electronic components, which contains 4 defective components, is rejected. A quality-control engineer randomly selects 4 components, and if 1 or more of them is defective, the shipment is rejected.

To find the probability of accepting the shipment, we need to calculate the probability of drawing 0 defective components from the sample of 4. This can be expressed using the hypergeometric distribution:

\[ P(X = 0) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}} = \frac{\binom{4}{0} \binom{156}{4}}{\binom{160}{4}} \]

Calculating this gives us:

\[ P(X = 0) = 0.9028 \]

The probability of rejecting the shipment is the complement of the probability of acceptance. Therefore, we can calculate it as follows:

\[ P(\text{Reject}) = 1 - P(X = 0) = 1 - 0.9028 = 0.0972 \]

Final Answer