Why is there a strong correlation between random number and correct guesses?

In trying to test whether knowing the history of a random number could help predict the future results, I found a strong, unexpected correlation between the average of the number generated, and the number of correct guesses.

The test was supposed to simulate flipping a coin (heads = 0, tails = 1) and if previous attempts were biased towards heads then guess tails and vice versa.

Why is the sum of the generated numbers always nearly equal to the number of correct guesses in the following LinqPad program?

void Main()
{
  var rnd = new Random();
  var attempts = 10000000;
  var correctGuesses = 0;
  long sum = 0;
  decimal avg = 0.5m;

  for (int i = 0; i < attempts; i++)
  {
    var guess = avg < 0.5m ? 1 : 0;
    var result = rnd.Next(0, 2);
    if (guess == result)
    {
      correctGuesses += 1;
    }
    sum += result;
    avg = (decimal)sum/(decimal)attempts;
  }

  attempts.Dump("Attempts");    
  correctGuesses.Dump("Correct Guesses");

  avg = (decimal)sum / (decimal)attempts;
  avg.Dump("Random Number Average");
}

Have a made an error in the code? Is this a natural relationship? I expected the averages to converge at 0.5 as I increased the number of attempts because the distribution is fairly even - I tested this with 10bn calls to Random.Next(0,2) - but I did not expect the sum of generated numbers to correlate to the number of correct guesses.