Let $n$ be a natural number. The leader of the math team invites $n$ girls for winter training, and each leaves her two gloves in a common box upon entry. The mischievous little brother randomly pairs the gloves into pairs, where each pair consists of one left glove and one right glove. A pairing is called weak if there is a set of $k < \frac{n}{2}$ pairs containing gloves of exactly $k$ girls. Find the probability that the pairing is not weak.