Acceptance sampling and quality control

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Problem 82 from Cacoullos T. "Exercises in Probability" [Cac89]More problems in Chapter 2 "Distributions. Random Variables"

A machine normally makes items of which $4\%$ are defective. Every hour the producer draws a sample of size 10 for inspection. If the sample contains no defective items he does not stop the machine. What is the probability that the machine will not be stopped when it has started producing items of which $10\%$ are defective.

Problem 158 from Cacoullos T. "Exercises in Probability" [Cac89]More problems in Chapter 4 "General Problems"

(a)(b)

Electronic tubes come in packages, each containing $N$ tubes. Let $p_{k}$ denote the probability that a package contains $k$ defective tubes ( $0 \leq k \leq m$ ). A sample of $n$ tubes is taken from a package and it is observed that $r \leq m$ tubes are defective.

(a)

What is the probability that the selected package actually contains $k$ ( $\geq r$ ) defective tubes?

(b)

If according to customer demand a package is considered not acceptable whenever it contains $d \geq r$ defective tubes, what is the probability that the package is not accepted?

Problem 162 from Cacoullos T. "Exercises in Probability" [Cac89]More problems in Chapter 4 "General Problems"

Quality control. In a continuous manufacturing process the percentage of defective articles is $p$ . To maintain the quality of the product at a certain standard, the articles are examined, one by one, until a sequence of fixed length $r$ appears with no defective article. Then total inspection terminates and only a certain fraction $f$ of the product is chosen at random for inspection until a defective article appears when the above process of $100\%$ inspection continues. Under the described sampling scheme find:

(a)

the probability of a defective sequence, i.e., that a defective article is not followed by $r$ successive good articles;

(b)

the expected number of articles in a defective sequence. Hence deduce the expected number, (i) of defective sequences, (ii) of inspected articles, after a defective article;

(c)

the expected proportion of the product subject to inspection;

(d)

the expected percentage $\rho$ of defective articles going out for sale provided that any discovered defective article is replaced by a good one;

(e)

for given values of $f$ and $r$ , what value $p^{*}$ , say, of $p$ maximizes $\rho$ .

Problem 7.10 from Smith G. "Exercises and Solutions in Probability and Statistics" [Smi26]More problems in Chapter 7 "Binomial Distribution"

In a 1979 court case, a military supplier argued that the US government had tested too small a sample when it rejected a shipment of 20,000 nose fuse adaptors, a component of artillery shells. Military procurement standards specified that a shipment of 10,001 to 35,000 adaptors would be tested by sampling 315 pieces, yet the government rejected this shipment of 20,000 pieces after testing 20 adapters and finding them all to be defective. If 20 items are randomly selected from an infinite population, in which $10 \%$ of the items are defective, what is the probability that all 20 items will be defective? That 15 or more will be defective?

References

[Cac89] Cacoullos T. Exercises in Probability. Springer-Verlag, 1989.

[Smi26] Smith G. Exercises and Solutions in Probability and Statistics. CRC Press, 2026.