Squeak: I think the real answer to your question is that the distribution of sample means approximates a normal distribution, but it does not do so precisely, and it gets more precise as the sample size grows bigger.
I used a sample size of 10 (I spun 10 times to divine what would happen if I spun it infinite times), which is not so large. I think if I ran it with a sample size of 100, it would be much closer to the mathematical answer.
The z score for that is an even 10, but I can’t find a z table that goes that low, so…