An Experimental Mathematics Perspective on the Old, and still Open, Question of When To Stop?

In a recent article in American Scientist, Theodore Hill described a coin-tossing game whose pay-off is the number of heads over the total number of throws. Suppose that at a given point during the game you have 5 heads and 3 tails, should you stop and get 5/8, or should you keep playing, hoping to get a better score? This is still an open problem. In the present article, we explore different strategies to this game from the Experimental Mathematics perspective.