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.
- (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. - 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.
- 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.
- 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.