Intersection, conditional probability and an independence test for two purchases, then an unknown probability found from independence

Two cafés, the Riverside café and the Station café, each sell muffins and fresh juice. For a customer chosen at random, let \(M\) be the event that the customer buys a muffin, \(J\) the event that the customer buys a juice, and \(N\) the event that the customer buys neither a muffin nor a juice.

At the Riverside café, \(\mathrm{P}(M)=0.42\), \(\mathrm{P}(J)=0.5\) and \(\mathrm{P}(N)=0.2\).

1. Show that \(\mathrm{P}(M\cap J)=0.12\).[3]
2. Find the probability that a customer at the Riverside café buys a muffin, given that the customer buys a juice.[3]
3. Determine whether \(M\) and \(J\) are independent events at the Riverside café. Justify your answer.[3]

At the Station café, \(\mathrm{P}(N)=0.32\). The probability that a customer buys a juice is three times the probability that the customer buys a muffin. At this café, buying a muffin is independent of buying a juice.

4. Find the probability that a customer at the Station café buys a muffin.[7]