Sum mutually exclusive winning rounds as a geometric series

Two players A and B each have three cards showing elephant, cat and mouse. They flash their cards simultaneously. Elephant defeats cat, cat defeats mouse, and mouse defeats elephant. Matching animals draw. In each round, A chooses elephant, cat and mouse with probabilities 0.3,0.3,0.4; B’s corresponding probabilities are 0.2,0.5,0.3. Play continues until there is a winner.

Find the probability that

(i) a round ends in a draw;

(ii) A wins the first round;

(iii) B wins the first round given that it is not a draw;

(iv) B wins the contest.