Multi-stage or conditional testing

3 The manufacturer of electronic components uses the following process to test the proportion of defective items produced. A random sample of 20 is taken from a large batch of components.

• If no defective item is found, the batch is accepted.
• If two or more defective items are found, the batch is rejected.
• If one defective item is found, a second random sample of 20 is taken. If two or more defective items are found in this second sample, the batch is rejected, otherwise the batch is accepted.

The proportion of defective items in the batch is denoted by \(p\), and \(q = 1 - p\).

1. Show that the probability that a batch is accepted is \(q ^ { 20 } + 20 p q ^ { 38 } ( q + 20 p )\). For a particular component, \(p = 0.01\).
2. Given that a batch is accepted, find the probability that it is accepted as a result of the first sample.

7 In a factory, an inspector checks a random sample of 30 mugs from a large batch and notes the number, \(X\), which are defective. He then deals with the batch as follows.

• If \(X < 2\), the batch is accepted.
• If \(X > 2\), the batch is rejected.
• If \(X = 2\), the inspector selects another random sample of only 15 mugs from the batch. If this second sample contains 1 or more defective mugs, the batch is rejected. Otherwise the batch is accepted.

It is given that \(5 \%\) of mugs are defective.

1. (a) Find the probability that the batch is rejected after just the first sample is checked.
   (b) Show that the probability that the batch is rejected is 0.327 , correct to 3 significant figures.
2. Batches are checked one after another. Find the probability that the first batch to be rejected is either the 4th or the 5th batch that is checked.
3. A bag contains 12 black discs, 10 white discs and 5 green discs. Three discs are drawn at random from the bag, without replacement. Find the probability that all three discs are of different colours.
4. A bag contains 30 red discs and 20 blue discs. A second bag contains 50 discs, each of which is either red or blue. A disc is drawn at random from each bag. The probability that these two discs are of different colours is 0.54 . Find the number of red discs that were in the second bag at the start.

4. A poultry farm produces eggs which are sold in boxes of 6 . The farmer believes that the proportion, \(p\), of eggs that are cracked when they are packed in the boxes is approximately 5\%. She decides to test the hypotheses $$\mathrm { H } _ { 0 } : p = 0.05 \text { against } \mathrm { H } _ { 1 } : p > 0.05$$ To test these hypotheses she randomly selects a box of eggs and rejects \(\mathrm { H } _ { 0 }\) if the box contains 2 or more eggs that are cracked. If the box contains 1 egg that is cracked, she randomly selects a second box of eggs and rejects \(\mathrm { H } _ { 0 }\) if it contains at least 1 egg that is cracked. If the first or the second box contains no cracked eggs, \(\mathrm { H } _ { 0 }\) is immediately accepted and no further boxes are sampled.

1. Show that the power function of this test is $$1 - ( 1 - p ) ^ { 6 } - 6 p ( 1 - p ) ^ { 11 }$$
2. Calculate the size of this test. Given that \(p = 0.1\)
3. find the expected number of eggs inspected each time this test is carried out, giving your answer correct to 3 significant figures,
4. calculate the probability of a Type II error. Given that \(p = 0.1\) is an unacceptably high value for the farmer,
5. use your answer from part (d) to comment on the farmer's test.