Let \(X\) be the number of years out of 15 that have New Year's Day on a Saturday. \(X\) follows a binomial distribution with parameters \(n = 15\) and \(p = \frac{1}{7}\).
We need to find \(P(X \geq 3)\), which is equivalent to \(1 - P(X \leq 2)\).
Calculate \(P(X = 0)\), \(P(X = 1)\), and \(P(X = 2)\):
\(P(X = 0) = \left( \frac{6}{7} \right)^{15}\)
\(P(X = 1) = \binom{15}{1} \left( \frac{1}{7} \right) \left( \frac{6}{7} \right)^{14}\)
\(P(X = 2) = \binom{15}{2} \left( \frac{1}{7} \right)^2 \left( \frac{6}{7} \right)^{13}\)
Thus, \(P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)\).
\(P(X \leq 2) = \left( \frac{6}{7} \right)^{15} + 15 \left( \frac{1}{7} \right) \left( \frac{6}{7} \right)^{14} + \binom{15}{2} \left( \frac{1}{7} \right)^2 \left( \frac{6}{7} \right)^{13}\)
\(P(X \leq 2) \approx 0.0990 + 0.2476 + 0.2889 = 0.6355\)
Therefore, \(P(X \geq 3) = 1 - 0.6355 = 0.3645\).
Thus, the probability that at least 3 of these years have New Year's Day on a Saturday is approximately 0.365 (accept 0.364).
