A lineup of \(100\) traders is ready to execute trades on the stock market. Each trader has a unique order tied to a specific stock ticker. For simplicity, let's say that the \(n\)-th trader in line has an order for stock ticker \(n\).
The first trader, overwhelmed by market noise and volatility, chooses to execute a trade on a random stock ticker (equally likely for each ticker). All the other traders are well-informed and will execute their trades on their specific stock tickers, unless it's already been traded. If the stock they intended to trade has already been traded, they will then choose another stock ticker to trade on, randomly.
What is the probability that the last trader will execute their trade on their designated stock ticker \((\#100)\)?