2020 ACSI EM S4 P1

There are \(2\) bags, \(A\) and \(B\). \(A\) contains \(3\) red balls and \(5\) green balls while \(B\) contains \(4\) red balls and \(3\) green balls. A game is played by \(2\) players where a fair \(6\) sided die is first thrown.

If the die shows the numbers \(1\) or \(2\), the player picks a ball from Bag \(A\) and places it in Bag \(B\). He then picks a ball from Bag \(B\). If the die shows \(3\), \(4\), \(5\) or \(6\), the player loses his turn. The other player will then throw the die. The player who picks a red ball from Bag \(B\) wins the game. Mark and Anthony take turns to play the game with Mark starting first. After each person has played the game, the balls are placed back into their original bags and the game starts again.

The game stops when a player has won.

Find the probability that

1. Mark wins the game on the first try.
2. Anthony wins the game on his first try.

Every ball selection is uniform at random. Each die throw is independent, and the bags are reset before the next turn if nobody has won.