I really liked this blog post on the St. Petersberg Paradox. It highlights something that is often overlooked when we think about randomness: the fact that system may not be ergodic, in the sense that the “average” behavior may not be the same if you are thinking about an average over space or an average over time. We usually think these are equivalent. for example, if you were a producer of a movie and you needed an establishing shot of an open ocean, you could take a helicopter and hover over one patch of water with your camera fixed (time average). If you were making an Indie film,and your budget didn’t allow for a helicopter, you could get a large photograph of the ocean and pan your camera over it (space average). Your audience would have a hard time distinguished these two cases. That means the process is ergodic. I first heard the word when I was introduced to Hillel Furstenberg (one of the only people I have spoken to on multiple occasions that has his own Wikipedia page). But non-ergodic randomness exists, and can create the seeming paradoxes if we are not ready for it.