Tree diagram for a spinner spun at most twice, the distribution and expectation of the total score, and the chance of a prize in two games

A fair spinner is divided into five equal sectors. Two sectors are green and three are white. Let \(G\) be the event that the spinner lands on green and \(W\) the event that it lands on white.

1. Write down
   1. \(\mathrm{P}(G)\);
   2. \(\mathrm{P}(W)\).[2]

The spinner is spun. When it lands on green, it is not spun again. When it lands on white, it is spun one more time. The tree diagram shows the possible outcomes, where \(u\), \(v\) and \(w\) are probabilities.

2. Find the values of \(u\), \(v\) and \(w\).[2]

In a game, each landing on green scores \(3\) points and each landing on white scores \(1\) point, following the spinning rule above. Let \(X\) be the total score in one game.

3. 1. Show that \(\mathrm{P}(X=4)=\dfrac{6}{25}\).
   2. Find \(\mathrm{P}(X=2)\).[3]
4. 1. Write down the probability distribution of \(X\) in a table.
   2. Find the expected total score.[5]
5. A player wins a prize when the total score is \(4\). The game is played twice. Find the probability that at least one prize is won.[4]