Output of code using System.Random does not approach theoretical limit as iterations increase

I'm not good at stats, so I tried to solve a simple problem in C#. The problem: "A given team has a 65% chance to win a single game against another team. What is the probability that they will win a best-of-5 set?"

I wanted to look at the relationship between that probability and the number of games in the set. How does a Bo3 compare to a Bo5, and so on?

I did this by creating Set and Game objects and running iterations. The win decision is done with this code:

Won = rnd.Next(1, 100) <= winChance;

rnd is, as you might expect, a static System.Random object.

Here's my Set object code:

public class Set
{
    public int NumberOfGames { get; private set; }
    public List<Game> Games { get; private set; }
    public Set(int numberOfGames, int winChancePct)
    {
        NumberOfGames = numberOfGames;
        GamesNeededToWin = Convert.ToInt32(Math.Ceiling(NumberOfGames / 2m));
        Games = Enumerable.Range(1, numberOfGames)
                          .Select(i => new Game(winChancePct))
                          .ToList();
    }

    public int GamesNeededToWin { get; private set; }
    public bool WonSet => Games.Count(g => g.Won) >= GamesNeededToWin;
}

My issue is that the results I get aren't quite what they should be. Someone who sucks less at stats did the math for me, and it seems my code is always overestimating the chance of winning the set, and the number of iterations doesn't improve the accuracy.

The results I get (% set win by games per set) are below. The first column is the games per set, the next is the statistical win rate (which my results should be approaching), and the remaining columns are my results based on the number of iterations. As you can see, more iterations don't seem to be making the numbers more accurate.

Games Per Set|Expected Set Win Rate|10K|100K|1M|10M

1 65.0% 66.0% 65.6% 65.7% 65.7%

3 71.8% 72.5% 72.7% 72.7% 72.7%

5 76.5% 78.6% 77.4% 77.5% 77.5%

7 80.0% 80.7% 81.2% 81.0% 81.1%

9 82.8% 84.1% 83.9% 83.9% 83.9%

The entire project is posted on github here if you want to look.

Any insight into why this isn't producing accurate results would be greatly appreciated.