To find the probability that at least one item will have a defect when 8 items are selected at random from a manufacturing process with a 3% defect rate, we can use the complement rule.
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Define the scenario:
- Probability of an item being defective (p) = 0.03
- Probability of an item being non-defective (q) = 1 - p = 1 - 0.03 = 0.97
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Calculate the probability that all 8 items are non-defective: The probability that all 8 chosen items are non-defective is given by: \[ P(\text{all non-defective}) = q^n = 0.97^8 \] where \( n = 8 \).
Now, we will calculate \( 0.97^8 \): \[ 0.97^8 \approx 0.783 \]
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Calculate the probability of having at least one defective item: The probability that at least one item is defective is the complement of all items being non-defective: \[ P(\text{at least one defective}) = 1 - P(\text{all non-defective}) \] \[ P(\text{at least one defective}) = 1 - 0.97^8 \approx 1 - 0.783 \approx 0.217 \]
Therefore, the probability that at least one of the 8 randomly chosen items will have a defect is approximately 0.217, or 21.7%.
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