(a) The tree diagram is constructed as follows:

• First level: Modes of transport with probabilities: Car (0.2), Bus (0.45), Walk (0.35).
• Second level: Arrival times for each mode: Early (E) or Not Early (NE).
• Probabilities for Car: E (0.6), NE (0.4).
• Probabilities for Bus: E (0.1), NE (0.9).
• Probabilities for Walk: E (1), NE (0).

(b) To find the probability that Juan goes to college by car given that he arrives early, we use Bayes' theorem:

\(P(C|E) = \frac{P(C \cap E)}{P(E)}\)

Where:

• \(P(C \cap E) = 0.2 \times 0.6 = 0.12\)
• \(P(E) = (0.2 \times 0.6) + (0.45 \times 0.1) + (0.35 \times 1) = 0.12 + 0.045 + 0.35 = 0.515\)

Thus,

\(P(C|E) = \frac{0.12}{0.515} = \frac{12}{515} \approx 0.233\)