5 A game is played with an ordinary fair 6-sided die. A player throws the die once. If the result is \(2,3,4\) or 5 , that result is the player's score and the player does not throw the die again. If the result is 1 or 6 , the player throws the die a second time and the player's score is the sum of the two numbers from the two throws.

1. Draw a fully labelled tree diagram to represent this information. Events \(A\) and \(B\) are defined as follows.
   \(A\) : the player's score is \(5,6,7,8\) or 9
   \(B\) : the player has two throws
2. Show that \(\mathrm { P } ( A ) = \frac { 1 } { 3 }\).
3. Determine whether or not events \(A\) and \(B\) are independent.
4. Find \(\mathrm { P } \left( B \mid A ^ { \prime } \right)\).