A factory has three machines A, B, C producing large numbers of a certain item. Of the total daily production of the item, 50% are produced on A, 30% on B and 20% on C. Records show that 2% of items produced on A are defective, 3% of items produced on B are defective and 4% of items produced on C are defective. The occurrence of a defective item is independent of all other items. One item is chosen at random from a day's total output.

(a) Show that the probability of its being defective is 0.027.

(b) Given that it is defective, find the probability that it was produced on machine A. (W)