(a) The tree diagram is completed with the following probabilities:

• 1 April Fine to 2 April Fine: 0.75
• 1 April Fine to 2 April Rainy: 0.25
• 1 April Rainy to 2 April Fine: 0.4
• 1 April Rainy to 2 April Rainy: 0.6

(b) The probability that 2 April is fine is calculated as follows:

\(P(\text{2 April Fine}) = 0.8 \times 0.75 + 0.2 \times 0.4 = 0.6 + 0.08 = 0.68\)

Thus, \(P(\text{2 April Fine}) = \frac{17}{25}\).

(c) To find \(P(X \cap Y)\), we calculate:

\(P(X \cap Y) = 0.8 \times 0.25 + 0.8 \times 0.6 = 0.2 + 0.12 = 0.27\)

(d) To find the probability that 1 April is fine given that 3 April is rainy, we use:

\(P(Y) = 0.2 \times 0.4 \times 0.25 + 0.2 \times 0.6 \times 0.6 = 0.362\)

\(P(X|Y) = \frac{P(X \cap Y)}{P(Y)} = \frac{0.27}{0.362} = 0.746\)