I have occasional discourses on probability. Here is Professor Munger having one upon the recent event that the Michigan Daily 4 lottery drew the same number two drawings in a row. The odds of that happening, of course, are exactly the same odds of your winning or of any given number being drawn once: 1/10,000. (If your intuition tells you it should be 1/100,000,000, remember that there are 10,000 ways it could happen.)
I haven’t checked his math on the all-year, many-states extension, but that’s the next piece on which to train your intuition: 1/10,000 events that have a chance to happen many times per day should happen pretty frequently. If you want the extended version of that, the post links to a piece adapted from a book subtitled, “Why Coincidences, Miracles, and Rare Events Happen Every Day.” Because in a world of seven billion people, one-in-a-million events happen seven thousand times a day.
Big, foundational ideas in probability theory were based on analysis of lottery and dice games like this. It is perhaps no wonder that we have trouble with designed more complicated games if our intuition has trouble with something as simple as drawing numbers from a hat.