In a large batch of items from a production line the probability that an item is faulty is p. 400 samples, each of size 5, are taken and the number of faulty items in each batch is noted. From the frequency distribution below estimate p and work out the expected frequencies of 0, 1, 2, 3, 4, 5 faulty items per batch for a theoretical binomial distribution having the same mean.

(a) Calculate, to two significant figures, the probability that, in any one sample, two bolts or less will be faulty,

(b) Find the expected value and the variance of the number of bolts in a sample which will (L. Additional) not be faulty.