(i) The mean number of cracked eggs is 1.4, so the probability of a cracked egg, p, is given by:

\(p = \frac{1.4}{20} = 0.07\)

The probability of exactly 2 cracked eggs in a box of 20 is calculated using the binomial distribution:

\(P(2) = \binom{20}{2} (0.07)^2 (0.93)^{18}\)

Calculating this gives:

\(P(2) = 0.252\)

(ii) The probability of at least 1 cracked egg is:

\(P(\text{at least 1 cracked egg}) = 1 - P(0)\)

Where:

\(P(0) = (0.93)^{20}\)

Thus:

\(P(\text{at least 1 cracked egg}) = 1 - (0.93)^{20} = 0.766\)

(iii) We need to find the smallest n such that:

\((0.766)^n < 0.01\)

Taking logarithms:

\(n \log(0.766) < \log(0.01)\)

Solving for n gives:

\(n > \frac{\log(0.01)}{\log(0.766)} \approx 17.8\)

Thus, the smallest integer n is 18.