[14 POINTS] Anne and Elizabeth like to challenge each other to competitions in cycling or swimming. For any randomly chosen competition, the probability that the competition is cycling is 0.5. When the competition is cycling Elizabeth wins with probability 0.9, whereas when the competition is swimming Elizabeth wins with probability 0.6. There are no ties or draw, so either Elizabeth or Anne can be the winner at each competition. (i) Are the events "The competition is cycling" and "the competition is swimming" independent? (ii) What is the probability that Elizabeth wins any random competition? (iii) If Anne has won the competition, what is the probability that it was a swimming competition