(i) The sum of all probabilities must equal 1:

\(6p + 0.1 = 1\)

Solving for \(p\):

\(6p = 0.9\)

\(p = 0.15\)

(ii) The variance \(\text{Var}(X)\) is calculated using:

\(\text{Var}(X) = (-1)^2 \times p + 0^2 \times p + 1^2 \times 2p + 2^2 \times 2p + 4^2 \times 0.1 - (1.15)^2\)

Substitute \(p = 0.15\):

\(\text{Var}(X) = 1 \times 0.15 + 0 + 1 \times 0.3 + 4 \times 0.3 + 16 \times 0.1 - 1.15^2\)

\(= 0.15 + 0 + 0.3 + 1.2 + 1.6 - 1.3225\)

\(= 1.9275\)

Rounded to three significant figures:

\(\text{Var}(X) = 1.93\)