In a school canteen, students can choose from a main course of meat $$\((M)\)$$, fish $$\((F)\)$$ or vegetarian $$\((V)\)$$. They can then choose a drink of either water $$\((W)\)$$ or juice $$\((J)\)$$. The partially completed tree diagram, where $$\(p\)$$ and $$\(q\)$$ are probabilities, shows the probabilities of these choices for a randomly selected student. (a) Complete the tree diagram, giving your answers in terms of $$\(p\)$$ and $$\(q\)$$ where appropriate. (2) (b) Find an expression, in terms of $$\(p\)$$ and $$\(q\)$$, for the probability that a randomly selected student chooses water to drink. (1) The events "choosing a vegetarian main course" and "choosing water to drink" are independent. (c) Find a linear equation in terms of $$\(p\)$$ and $$\(q\)$$. (2) A student who has chosen juice to drink is selected at random. The probability that they chose fish for their main course is $$\(\frac{7}{30}\)$$ (d) Find the value of $$\(p\)$$ and the value of $$\(q\)$$. (5) The canteen manager claims that students who choose water to drink are most likely to choose a fish main course. (e) State, showing your working clearly, whether or not the manager's claim is correct. (3)

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