(i) To find the probability distribution, consider all possible outcomes when two dice are thrown. There are 36 possible outcomes (6 sides on the first die and 6 sides on the second die).

- The score is 0 when both dice show the same number. There are 6 such outcomes: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6). So, P(0) = \(\frac{6}{36}\).

- For scores 1 to 5, count the outcomes where the smaller number is 1, 2, 3, 4, or 5, respectively:

• Score 1: (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (3,1), (4,1), (5,1), (6,1) - 10 outcomes, so P(1) = \(\frac{10}{36}\).
• Score 2: (2,3), (2,4), (2,5), (2,6), (3,2), (4,2), (5,2), (6,2) - 8 outcomes, so P(2) = \(\frac{8}{36}\).
• Score 3: (3,4), (3,5), (3,6), (4,3), (5,3), (6,3) - 6 outcomes, so P(3) = \(\frac{6}{36}\).
• Score 4: (4,5), (4,6), (5,4), (6,4) - 4 outcomes, so P(4) = \(\frac{4}{36}\).
• Score 5: (5,6), (6,5) - 2 outcomes, so P(5) = \(\frac{2}{36}\).

(ii) The expected score is calculated using the formula for expected value: \(E(X) = \sum x_i p_i\).

\(E(X) = 0 \times \frac{6}{36} + 1 \times \frac{10}{36} + 2 \times \frac{8}{36} + 3 \times \frac{6}{36} + 4 \times \frac{4}{36} + 5 \times \frac{2}{36}\)

\(E(X) = \frac{0 + 10 + 16 + 18 + 16 + 10}{36} = \frac{70}{36} \approx 1.94\)