1. The bag \(P\) contains 6 balls of which 3 are red and 3 are yellow.

The bag \(Q\) contains 7 balls of which 4 are red and 3 are yellow.
A ball is drawn at random from bag \(P\) and placed in bag \(Q\). A second ball is drawn at random from bag \(P\) and placed in bag \(Q\).
A third ball is then drawn at random from the 9 balls in bag \(Q\). The event \(A\) occurs when the 2 balls drawn from bag \(P\) are of the same colour. The event \(B\) occurs when the ball drawn from bag \(Q\) is red.

1. Complete the tree diagram shown below.
   (4)
   \includegraphics[max width=\textwidth, alt={}, center]{c78ec7b6-dd06-4de1-94c2-052a5577dd10-12_1201_1390_753_269}
2. Find \(\mathrm { P } ( A )\)
3. Show that \(\mathrm { P } ( B ) = \frac { 5 } { 9 }\)
4. Show that \(\mathrm { P } ( A \cap B ) = \frac { 2 } { 9 }\)
5. Hence find \(\mathrm { P } ( A \cup B )\)
6. Given that all three balls drawn are the same colour, find the probability that they are all red.
   (3)