The basic idea is that if you toss a coin many times, say 100, you should expect to get about the same number of heads and tails. If you tossed it many more times than 100, like 10,000, you would expect to get even closer to a 1:1 ratio of heads and tails (closer and closer to 50% heads and 50% tails). The more times you repeat the experiment, the closer you should get to the true theoretical average of heads and tails.
It made me think about the number of types of events or encounters which occur in any particular game, and the ratio between them and the number of turns it takes to play through a game. As the number of turns to play increases and the number of types of events decreases, you get a more even, predictable, and possibly boring experience in terms of variety, but as the number of turns to play decreases and/or the number of types of events increases, you get a less predictable, more chaotic experience.
So if your game seems repetitive and boring, you might experiment with decreasing the length of it, or increasing the number of types of things you do in it. On the other hand, if your game seems too wild and crazy or overwhelming, you might experiment with increasing the length of it, or decreasing the types of things you do in it.
Undoubtedly there are many other ways to think about using the law of large numbers and probability to understand game design better.
