(i) The number of fireworks that fail follows a binomial distribution with parameters \(n = 20\) and \(p = 0.05\). We need to find \(P(X > 1)\).

\(P(X > 1) = 1 - P(X = 0) - P(X = 1)\)

\(P(X = 0) = (0.95)^{20}\)

\(P(X = 1) = \binom{20}{1} (0.95)^{19} (0.05)^1\)

\(P(X > 1) = 1 - (0.95)^{20} - \binom{20}{1} (0.95)^{19} (0.05)\)

\(P(X > 1) = 1 - 0.3585 - 0.3770 = 0.264\)

(ii) The profit if 19 or 20 fireworks work is \(450 \times 10 - 480 = 4020\).

If more than 1 firework fails, the profit is \(-480\).

The expected profit is:

\(4020 \times (1 - 0.264) - 480 \times 0.264\)

\(= 4020 \times 0.736 - 480 \times 0.264\)

\(= 2960.64 - 126.72 = 2833.92\)

Rounding gives \(\$2830\).