Answer :
The probability that a particular shipment (with nine percent defectives) will be accepted as good anyway is 0.0474
Probability of defective item in the shipment, p = 0.09
The policy is to return the shipment if 5% or more are defective. If a sample of 100 is selected, in order to get it accepted not more than 4 items should be found defective.
Calculate the required probability.
P=
{¹⁰⁰C₀ × p⁰ x (1 - p)¹⁰⁰+¹⁰⁰C₁ × p¹× (1 - p)⁹⁹ +¹⁰⁰C₂ × p² ×(1 − p)⁹⁸ + ¹⁰⁰C₃ × p³ ×(1 − p)⁹⁷+¹⁰⁰C₄ × p⁴×(1 − p)⁹⁶ }
Substitute the values.
P = {¹⁰⁰C₀ × 0.09⁰ x (1 - 0.09)¹⁰⁰+¹⁰⁰C₁ × 0.09¹× (1 - 0.09)⁹⁹ +¹⁰⁰C₂ × 0.09² ×(1 − 0.09)⁹⁸ + ¹⁰⁰C₃ × 0.09³ ×(1 − 0.09)⁹⁷+¹⁰⁰C₄ × 0.09⁴×(1 − 0.09)⁹⁶ }
= {¹⁰⁰C₀ × 0.09⁰ x (0.91)¹⁰⁰+¹⁰⁰C₁ × 0.09¹× (0.91)⁹⁹ +¹⁰⁰C₂ × 0.09² ×(0.91)⁹⁸ + ¹⁰⁰C₃ × 0.09³ ×(0.91)⁹⁷+¹⁰⁰C₄ × 0.09⁴×(0.91)⁹⁶ }
= 0.0474
Hence, the probability that a particular shipment (with nine percent defectives) will be accepted as good anyway is 0.0474.
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