(i) To find \(P(X = 2)\), consider the cases:

- If Gohan throws a 1, she must throw again and get a 1 to score 2. The probability is \(\frac{1}{4} \times \frac{1}{4} = \frac{1}{16}\).

- If Gohan throws a 2, her score is 2. The probability is \(\frac{1}{4}\).

Thus, \(P(X = 2) = \frac{1}{16} + \frac{1}{4} = \frac{5}{16}\).

(ii) To calculate \(E(X)\):

\(E(X) = \sum xP(X = x) = 2 \times \frac{5}{16} + 3 \times \frac{1}{16} + 4 \times \frac{3}{8} + 5 \times \frac{1}{8} + 6 \times \frac{1}{16} + 7 \times \frac{1}{16}\)

\(= \frac{10}{16} + \frac{3}{16} + \frac{12}{16} + \frac{10}{16} + \frac{6}{16} + \frac{7}{16} = \frac{60}{16} = 3.75\).

To calculate \(\text{Var}(X)\):

\(\text{Var}(X) = \sum x^2 P(X = x) - (E(X))^2\)

\(= 2^2 \times \frac{5}{16} + 3^2 \times \frac{1}{16} + 4^2 \times \frac{3}{8} + 5^2 \times \frac{1}{8} + 6^2 \times \frac{1}{16} + 7^2 \times \frac{1}{16} - (3.75)^2\)

\(= \frac{20}{16} + \frac{9}{16} + \frac{48}{16} + \frac{25}{8} + \frac{36}{16} + \frac{49}{16} - 14.0625\)

\(= \frac{260}{16} - \frac{225}{16} = \frac{35}{16} = 2.19\).