(i) We use the binomial probability formula:

\(P(X = r) = \binom{n}{r} p^r (1-p)^{n-r}\)

where \(n = 12\), \(p = 0.65\), and \(1-p = 0.35\).

Calculate for \(r = 8, 9, 10\):

\(\binom{12}{8} (0.65)^8 (0.35)^4 + \binom{12}{9} (0.65)^9 (0.35)^3 + \binom{12}{10} (0.65)^{10} (0.35)^2\)

\(= 0.541\)

(ii) The probability that the fourth passenger is the first to carry a backpack is calculated by considering the first three passengers do not carry a backpack and the fourth does:

\(P(\overline{R}\overline{R}\overline{R}R) = (0.35)^3 \times 0.65\)

\(= 0.0279\)