The waiting time paradox: why is my bus always late? (2018)

Instead of 1 in 14 million, we'll just do 1 in 2, and 8 players.

So we'll check how many bits are set in the average random byte:

    void Main()
     {
      byte[] buffer = new byte[256*1024];
      Random.Shared.NextBytes(buffer);
      var avg = buffer.Average(b =>     System.Runtime.Intrinsics.X86.Popcnt.PopCount(b));
      Console.WriteLine(avg);
     }

Now let's check how many bits are set in the average random byte given that the low bit is 1 (i.e. player 1 has won!)

    void Main()
     {
      byte[] buffer = new byte[256*1024];
      Random.Shared.NextBytes(buffer);
      var avg = buffer
       .Where(b => (b & ((byte)0x01)) == 0x01)
       .Average(b =>     System.Runtime.Intrinsics.X86.Popcnt.PopCount(b));
      Console.WriteLine(avg); 
     }

Which is 1+3.5

In this case, we're 1+ average from the 7 other players, so being an average of 7 others not 8 others is significant.

If we simulate with millions of players, you'll see that removing 1 person from the pool makes essentially no difference.