2008 JJC P2 Q7

In a game played by two people A and B, each player is given three cards with three different animal pictures printed on them. The players flash their cards simultaneously to indicate one of the three animals, ‘elephant’, ‘cat’, and ‘mouse’. ‘Elephant’ defeats ‘cat’, ‘cat’ defeats ‘mouse’ and ‘mouse’ defeats ‘elephant’. If the animals are the same, the round is a draw. At each round, A indicates ‘elephant’, ‘cat’, and ‘mouse’ with probabilities \(0.3\), \(0.3\) and \(0.4\) respectively, while the corresponding probabilities for B are \(0.2\), \(0.5\) and \(0.3\). The game continues until a winner is found. Find the probability that

1. a round ends in a draw,
2. A is the winner in the first round,
3. B is the winner in the first round given that the first round does not end in a draw,
4. B will be the winner in the contest.