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A factory produces items, each of which is defective with probability p. From a lot, a sample of n items is taken with replacement. If the sample contains no defective item, the lot is accepted; if it contains more than two defective items, the lot is rejected. If the sample contains exactly one or two defective items, then another independent sample of n items is drawn with replacement from the same lot and added to the first sample. If the total combined sample still has at most two defective items, the lot is accepted. The probability that the lot is accepted is

1. qⁿ + npq^(n-1) (q^m + mpq^(m-1)) + n(n-1)/2 p² q^(n+m-2)
2. npq^(n-1) (q^m + mpq^(m-1))
3. qⁿ + n(n-1)/2 p² q^(n+m-2)
4. None of these

Correct answer: qⁿ + npq^(n-1) (q^m + mpq^(m-1)) + n(n-1)/2 p² q^(n+m-2)

Solution

Accept if first sample has 0 defects (q^n); if exactly 1 defect (npq^(n-1)) then second sample needs <=1 defect (q^m+mpq^(m-1)); if exactly 2 defects (n(n-1)/2 p^2 q^(n-2)) then second sample needs 0 defects (q^m). Total = q^n + npq^(n-1)(q^m+mpq^(m-1)) + n(n-1)/2 p^2 q^(n+m-2).

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