But what happens to the grey box that represents our upper and lower confidence bounds? It will shrink every time we play a round. This is because the more rounds we play, the more accurate our observed average will become and, therefore, the narrower our confidence bounds will be. The size of this box, then, will be inversely proportional to the number of rounds we have played on that machine. The smaller this box is, the more confident we can be that we are getting closer to the machine’s true expected return. This is a direct consequence of the law of large numbers.
