(a) A factory makes bottles of juice using only two machines, A and B. The volume of the juice produced by them are normally distributed. Machine A produces bottles of juice with a mean volume of 503ml, and a variance of 9ml2; while machine B produces bottles with a mean volume of 498ml, and a variance of 4m12. It is also known that 30% of the output of the factory has a volume less than 500ml. Let A be the event that the bottle was produced by machine A and B be the event that the bottle was produced by machine B, and C be the event that the chosen bottle has volume less than 500ml. (i) If a bottle of juice is produced by machine A, what is the probability that it has a volume less than 500 ml? [4 marks] (ii) Find the probability that a randomly selected bottle is produced by machine B. [4 marks] (iii) Are events A and C independent? [2 marks]