Read a probability tree, calculate conditional probabilities and form an expected-cost distribution.

José travels to school on a bus. On any day, the probability that José will miss the bus is \(\frac13\). If he misses his bus, the probability that he will be late for school is \(\frac78\). If he does not miss his bus, the probability that he will be late is \(\frac38\). Let \(E\) be the event “he misses his bus” and \(F\) the event “he is late for school”. The information is shown on the following tree diagram.

1. Find[4]
   1. \(P(E ∩ F)\);
   2. \(P(F)\).
2. Find the probability that[5]
   1. José misses his bus and is not late for school;

   2. José missed his bus, given that he is late for school. The cost for each day that José catches the bus is \(3\) euros. José goes to school on Monday and Tuesday.

3. Copy and complete the probability distribution table.[3]

   \(X\) (cost in euros)	\(0\)	\(3\)	\(6\)
   \(P(X)\)	\(\frac{1}{9}\)

4. Find the expected cost for José for both days.[2]