A particular industrial product is shipped in lots of 20. Testing to determine whether an item is defective is costly; hence, the manufacturer samples production rather than using a 100% inspection plan. A sampling plan constructed to minimize the number of defectives shipped to customers calls for sampling five items from each lot and rejecting the lot if more than one defective is observed. (If the lot is rejected, each item in the lot is then tested.) If a lot contains four defectives, what is the probability that it will be accepted?
Answers
Given product is shipped in lots of 20
sampling five items from each lot and rejecting the lot if more than one defective is observed.
a lot contains four defectives,
To Find : the probability that lot will be accepted
Solution:
Lot size = 20
Defectives = 4
Sample Size = 5
5 Samples out of 20 can be selected in ²⁰C₅ ways
Lots will be accepted if 0 sample is selected from 4 defectives and 5 samples selected from 16 non defectives or 1 sample is selected from 4 defectives and 4 samples selected from 16 non defectives
Hence number of ways = ⁴C₀*¹⁶C₅ + ⁴C₁*¹⁶C₄ = ¹⁶C₅ + 4 * ¹⁶C₄
probability that lot will be accepted = (¹⁶C₅ + 4 * ¹⁶C₄ )/ ²⁰C₅
= ( 4,368 + 7,280)/(15,504)
= (11,648 / 15,504)
= 0.7513
the probability that lot will be accepted = 0.7513
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