You are discussing the way brain could produce chance consciously, and I played your game for a while, but in fact, the random process that I am talking about would be subconscious. For instance, when we suddenly realize that we are having a good intuition, in reality, what we would have is a good feeling about an idea that seems to come from nowhere, which would be the case if it had developed randomly. To me, our intuitions show how our ideas could develop randomly, whereas our improvisations show how our actions could do so. These two phenomenon concern the future, they only work if we have time to check them out.
When we have a good intuition, we can build an hypothesis out of it, and then make an experiment to see if it works. And when we improvise, we can make random moves, but it will work only if we have time to catch up with the imprecision it adds up to our actions. For instance, a Jazz player can make false notes only if he is able to catch up with the theme and the rhythm later, and the more he takes chances, the more his music will sound good to us if he is still able to catch up.