2022 SAJC P2 Q8

In a game, two boxes have the following contents.

Box \( \; A \; \) contains \( \; 3 \; \) green balls and \( \; 4 \; \) white balls.

Box \( \; B \; \) contains \( \; 6 \; \) green balls and \( \; 5 \; \) white balls.

A fair die is thrown once. Two balls are then drawn in sequence in the following manner:

1. If the number that appears on the top face of the die is less than \(3\), the first ball is drawn from Box \(A\) and transferred to Box \(B\). Otherwise, the first ball is drawn from Box \(B\) and transferred to Box \(A\).

2. The second ball is then randomly picked from the box that received the transferred ball in step \(1\).

1. Draw a probability tree diagram to represent the above information.[2]
2. If the second ball drawn is white, the player wins the game. Otherwise, the player loses the game. Find the probability that the first ball drawn is from Box \(A\), given that the player wins the game.[5]