To find the probability that at least one item has a defect when 9 items are chosen at random from a manufacturing machine with a 10% defect rate, we can use the complementary probability approach.
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Define the parameters:
- Let \( p \) be the probability that an item is defective (which is 0.10 or 10%).
- Let \( q \) be the probability that an item is not defective (which is \( 1 - p = 0.90 \) or 90%).
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Calculate the complementary probability:
- The complementary event to "at least one defect" is "no defects". To find the probability of no defects in 9 items, you would calculate \( q^n \) where \( n \) is the number of items (in this case, 9).
\[ P(\text{no defects}) = q^9 = 0.90^9 \]
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Calculate \( 0.90^9 \): \[ 0.90^9 \approx 0.3874 \]
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Calculate the probability of at least one defect: \[ P(\text{at least one defect}) = 1 - P(\text{no defects}) = 1 - 0.90^9 \] \[ P(\text{at least one defect}) \approx 1 - 0.3874 \approx 0.6126 \]
Therefore, the probability that at least one of the 9 items is defective is approximately \( 0.6126 \), or 61.26%.
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