Use set probabilities, conditional probability and independence in two restaurant models.

Two restaurants, Center and New, sell fish rolls and salads. Let \(F\) be the event a customer chooses a fish roll. Let \(S\) be the event a customer chooses a salad. Let \(N\) be the event a customer chooses neither a fish roll nor a salad. In the Center restaurant \(P(F) = 0.31\), \(P(S) = 0.62\), \(P(N) = 0.14\).

1. Show that \(P(F ∩ S) = 0.07\).

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2. Given that a customer chooses a salad, find the probability the customer also chooses a fish roll.[3]

3. Are \(F\) and \(S\) independent events? Justify your answer. At New restaurant, \(P(N) = 0.14\). Twice as many customers choose a salad as choose a fish roll. Choosing a fish roll is independent of choosing a salad.

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4. Find the probability that a fish roll is chosen.[7]