(i) The number of eggs laid by the hens follows a binomial distribution with parameters \(n = 30\) and \(p = 0.7\). We need to find \(P(X = 24)\).

Using the binomial probability formula:

\(P(X = 24) = \binom{30}{24} (0.7)^{24} (0.3)^{6}\)

\(= (0.7)^{24} \times (0.3)^{6} \times \binom{30}{24}\)

\(= 0.0829\)

(ii) To approximate the probability that fewer than 20 eggs are laid, we use a normal approximation to the binomial distribution.

The mean \(\mu\) and variance \(\sigma^2\) of the binomial distribution are given by:

\(\mu = np = 30 \times 0.7 = 21\)

\(\sigma^2 = np(1-p) = 30 \times 0.7 \times 0.3 = 6.3\)

Using the normal approximation, we standardize the variable:

\(P(X < 20) \approx P\left(Z < \frac{19.5 - 21}{\sqrt{6.3}}\right)\)

\(= P(Z < -0.5976)\)

Using standard normal distribution tables:

\(P(Z < -0.5976) = 1 - 0.7251 = 0.275\)