As you do repeated trials, the mean average of those trials will approach the theoretical mean. Or if you are looking at a sample, as your sample grows, its mean average will get closer to the mean of the entire population.
The law of large numbers is a good title for this. Many have the idea that the law of probability would imply that a .250 hitter that goes 0 for 3 is now "due" for a hit. Flipping a coin 10 times means you'll get 5 heads and 5 tails. For most of us, our own life experience would show statements like these to be incorrect. It would not be weird for a coin being flipped 10 times to have 3 heads and 7 tails. However, we would think something was up if we flipped it 1,000 times and got 300 heads and 700 tails.
Like many math teachers, I would have classes do some coin flipping experiments. Always a fun day. For years I would write down the results and keep a running total. I don't know where that is now. I wish I had kept that. I was up to something like 20,000 flips. It wasn't 50-50, but pretty close. Maybe something like 49.7% to 50.3%.
I'm reminded of this in something I read in "The Signal and the Noise: Why So Many Predictions Fail - But Some Don't" by Nate Silver. It's an interesting book. One example: At what is a major league baseball player at his peak? You can make a pretty good case for it being 27 years of age. To make his case, he looked at 50 MVP award winners. Granted, 50 is possibly not to be considered a "large number", but it's what was used in this case.
"A baseball player...peaks at age twenty-seven. Of the fifty MVP winners between 1985 and 2009, 60 percent were between the ages of twenty-five and twenty-nine, and 20 percent were aged twenty-seven exactly."
This doesn't prove anything for sure, but then again, surveys never do. I would think we would have a better idea as we can look at additional MVP's in coming decades, giving more applicability to the law of large numbers. Also, the results might have been more convincing if they didn't include Barry Bonds steroid-assisted MVP awards in his late thirties.
