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What does the Law of Large Numbers really say?

Here are the cumulative results of the six sets of 200 real coin tosses you generated in class. (Each color corresponds to a different set. I concatenated the sets to make one long sequence of 1200 tosses). I counted a Head as +1 and a Tail as -1. The first graph shows the variable 'fortune', which is the cumulative sum of the +1's and -1's, as the tossing proceeds. The popular "Law of Averages" leads many people to think the fortune will come back to zero often due to some sort of compensation effect. Is this what you see?


 
The true "Law of Large Numbers" says something about what happens to the average fortune over the long run. Here is a graph of the average fortune as the tossing proceeds. Each point corresponds to the above fortune divided by the number of tosses seen so far. Now what do you see?