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A manufactured lot of buggy whips has 20 items, of which 5 are defective. A random sample of 5 items is chosen to be inspected. We need to find the probability that the sample contains exactly one defective item under two scenarios:

  1. Sampling is done with replacement.
  2. Sampling is done without replacement.

Solution

(a) Sampling with Replacement

When sampling with replacement, each item is put back into the lot before the next item is selected, meaning the probability of picking a defective or non-defective item remains constant with each selection.

  • (P(D)): \(\frac{1}{4}\)
  • (P(N)): \(\frac{3}{4}\)

The sample can contain exactly one defective item in any one of the following sequences: DNNNN, NDNNN, NNDNN, NNNDN, NNNND, where D represents a defective item and N represents a non-defective item.

the total probability is:

\[ P(\text{exactly one defective}) = 5 \times \left(\frac{1}{4}\right) \times \left(\frac{3}{4}\right)^4 = 0.396 \]

(b) Sampling without Replacement

When sampling without replacement, each selected item is not returned to the lot, affecting the probabilities with each selection.

\[ P(\text{exactly one defective}) = \frac{\binom{5}{1} \times \binom{15}{4}}{\binom{20}{5}} = 0.440 \]

Conclusion

  • With replacement: The probability is approximately 0.396.
  • Without replacement: The probability is approximately 0.440.