Let \(X\) be the number of years out of 56 where New Year's Day is on a Saturday. \(X\) follows a binomial distribution with parameters \(n = 56\) and \(p = \frac{1}{7}\).

We approximate this binomial distribution with a normal distribution. The mean \(\mu\) and variance \(\sigma^2\) are given by:

\(\mu = 56 \times \frac{1}{7} = 8\)

\(\sigma^2 = 56 \times \frac{1}{7} \times \frac{6}{7} = 6.857\)

We need to find \(P(X > 7)\). Using a continuity correction, this becomes \(P(X > 7.5)\).

Standardizing, we have:

\(P(X > 7.5) = 1 - \Phi \left( \frac{7.5 - 8}{\sqrt{6.857}} \right)\)

\(= 1 - \Phi(-0.1909)\)

\(= \Phi(0.1909)\)

Using standard normal distribution tables, \(\Phi(0.1909) \approx 0.576\).