(i) The probability that Amy loses her original $1 is the probability that she fails both attempts. The probability of failing one attempt is given by:

\(P(F) = 1 - 0.2 = 0.8\)

Therefore, the probability of failing both attempts is:

\(P(\text{loses } \$1) = P(F \text{ and } F) = 0.8 \times 0.8 = 0.64\)

(ii) The probability distribution table for the amount that Amy gains is:

The probability of gaining $0.50 is the probability of failing the first attempt and succeeding on the second:

\(P(0.50) = P(F \text{ and } S) = 0.8 \times 0.2 = 0.16\)

The probability of gaining $2 is the probability of succeeding on the first attempt:

\(P(2) = 0.2\)

(iii) The expected gain is calculated as follows:

\(E(\text{winnings}) = (-1 \times 0.64) + (0.5 \times 0.16) + (2 \times 0.2)\)

\(E(\text{winnings}) = -0.64 + 0.08 + 0.4 = -0.16\)

Therefore, Amy's expected gain is -$0.16, indicating a loss of 16 cents.