(i) The tree diagram is as follows:

• First branch: Economy (E) with probability 0.82, First class (F) with probability 0.18.
• Second branch from Economy: Good Night's Sleep (GNS) with probability \(x\), Not GNS with probability \(1-x\).
• Second branch from First class: GNS with probability 0.9, Not GNS with probability 0.1.

(ii) The probability of getting a good night's sleep is given by:

\(0.82x + 0.18 imes 0.9 = 0.285\)

Solving for \(x\):

\(0.82x + 0.162 = 0.285\)

\(0.82x = 0.123\)

\(x = \frac{0.123}{0.82} = 0.15\)

(iii) We need to find \(P(E | \text{not GNS})\):

\(P(E | \text{not GNS}) = \frac{P(E \cap \text{not GNS})}{P(\text{not GNS})}\)

\(P(E \cap \text{not GNS}) = 0.82 imes (1 - 0.15) = 0.82 imes 0.85 = 0.697\)

\(P(\text{not GNS}) = 1 - 0.285 = 0.715\)

\(P(E | \text{not GNS}) = \frac{0.697}{0.715} = 0.975\)