The problem involves a binomial distribution where the probability of success (a resident being in favour) is 0.8, and the number of trials is 20. We need to find the probability that more than 17 residents are in favour, i.e., \(P(X > 17)\).

Using the binomial probability formula:

\(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\)

We calculate:

\(P(X = 18) = \binom{20}{18} (0.8)^{18} (0.2)^2\)

\(P(X = 19) = \binom{20}{19} (0.8)^{19} (0.2)^1\)

\(P(X = 20) = \binom{20}{20} (0.8)^{20}\)

Summing these probabilities gives:

\(P(X > 17) = P(X = 18) + P(X = 19) + P(X = 20)\)

Calculating each term:

\(P(X = 18) = \binom{20}{18} (0.8)^{18} (0.2)^2 = 0.13691\)

\(P(X = 19) = \binom{20}{19} (0.8)^{19} (0.2)^1 = 0.05765\)

\(P(X = 20) = \binom{20}{20} (0.8)^{20} = 0.01153\)

Adding these gives:

\(P(X > 17) = 0.13691 + 0.05765 + 0.01153 = 0.20609\)

Rounding to three decimal places, the probability is 0.206.