(i) The tree diagram is as follows:

First attempt: Pass (0.85), Fail (0.15)

If Fail: Retake Pass (0.65), Retake Fail (0.35)

(ii) We need to find the probability that a student fails the first attempt given that they gain the certificate.

Let \(F\) be the event of failing the first attempt and \(P\) be the event of passing overall.

The probability of passing overall, \(P(P)\), is given by:

\(P(P) = P(\text{Pass first attempt}) + P(\text{Fail first attempt and Pass retake})\)

\(P(P) = 0.85 + (0.15 \times 0.65) = 0.85 + 0.0975 = 0.9475\)

The probability of failing the first attempt and passing the retake, \(P(F \cap P)\), is:

\(P(F \cap P) = 0.15 \times 0.65 = 0.0975\)

Using conditional probability:

\(P(F | P) = \frac{P(F \cap P)}{P(P)} = \frac{0.0975}{0.9475} = \frac{39}{379} \approx 0.103\)