Rapid-Test Kit Defects and Batch Return Decision
A university laboratory receives rapid-test kits from two suppliers. Supplier A provides 70% of the kits, and Supplier B provides 30%. Past quality checks show that 1% of Supplier A's kits and 8% of Supplier B's kits are defective. A kit is selected at random from a combined stock of 500 kits and tested. It is defective. The lab can either return the entire batch from the supplier that most likely produced the defective kit, or test every kit at a cost of 4 KD per kit. Returning a batch costs 150 KD in administrative fees and requires replacing all kits in that batch at 5 KD per kit. The lab director wants to know which supplier most likely produced the defective kit, the expected number of defective kits in the stock, and whether returning the likely batch is cheaper than testing all kits. The director also wants to know what defects would remain under the cheaper plan.
1. Compute the overall defect rate, the expected number of defective kits in 500, and the probability that the defective kit came from Supplier B.
Read the worked answer
The overall defect probability is
P(defective) = 0.70(0.01) + 0.30(0.08) = 0.007 + 0.024 = 0.031.
So the expected number of defective kits in 500 is 500(0.031) = 15.5.
By Bayes' rule,
P(Supplier B | defective) = 0.30(0.08) / 0.031 = 0.024 / 0.031 = 0.774, about 77.4%.
Thus Supplier B is the more likely source, but there is still about a 22.6% chance that the defective kit came from Supplier A.
2. Compare the immediate cost of returning the entire B batch with testing all 500 kits. Which is cheaper, and what important limitation does the cheaper option have?
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Supplier B's batch contains 0.30(500) = 150 kits. Returning that batch costs
150 KD administration + 150(5 KD) replacement = 150 + 750 = 900 KD.
Testing all 500 kits costs 500(4) = 2,000 KD. Returning the B batch is cheaper by 1,100 KD.
The limitation is that returning B does not remove expected defects from Supplier A. Supplier A's expected defective count is 350(0.01) = 3.5 kits. Also, because P(Supplier A | defective) is about 22.6%, there is a real chance the observed defective kit came from A. If the goal is to identify every defective kit, testing all 500 is the only option that does that.
3. If the lab instead tests only the 150 kits from Supplier B, what is the expected number of defective kits found and expected number missed from Supplier A? Does this plan meet a goal of finding all defective kits?
Read the worked answer
Testing only Supplier B finds an expected 150(0.08) = 12 defective B kits, at a testing cost of 150(4) = 600 KD. It does not test Supplier A, whose expected defective count is 350(0.01) = 3.5 kits.
So the plan finds expected 12 defective kits and misses expected 3.5 defective kits from A. It is cheaper than testing all 500 kits, but it does not meet a goal of finding all defective kits. It addresses the higher-rate supplier while leaving a smaller but real defect risk in the A batch.