2008 JJC P2 Q9

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 game is a draw. At each game, \(A\) indicates ‘elephant’, ‘cat’ and ‘mouse’ with probabilities of \(0.3\), \(0.3\) 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 game ends in a draw.[2]
2. \(A\) is the winner in the first game;[2]
3. \(B\) is the winner in the first game given that the first game does not end in a draw;[3]
4. \(B\) will be the winner in the contest.[2]