2015 VJC P2 Q7

A game is to be played between two players, \(A\) and \(B\). A bag contains \(5\) balls each with \(A\)’s name and \(8\) balls each with \(B\)’s name. Starting from \(A\), the players will take turn to pick \(3\) balls randomly in a single draw from the bag, note down the names and return all \(3\) balls to the bag. Assuming that the balls are identical in size, the winner is the first player to get \(3\) balls of the player’s name in a single draw.

1. Find the probability that \(A\) wins the game.[4]
2. Find the probability that \(A\) wins on \(A\)’s first draw given that \(A\) wins the game.[3]

Suppose \(A\) eventually wins the game on \(A\)’s \(4\)th turn. Let (\({{a}_{1}}\), \({{a}_{2}}\), \({{a}_{3}}\), \({{a}_{4}}\)) denote \(A\)’s draw sequence where \({{a}_{i}}\) denote the number of balls with \(A\)’s name picked by \(A\) on \(A\)’s \(i\)th turn.

Find the total number of possible draw sequence for \(A\).[2]