We take for granted many intuitions, but it's important to remember what intuitions are truly universal, and which are the function of explicit training. For example, speech is hard-wired into our brains, and children exposed to people talking will naturally learn to talk. In contrast, kids won't learn to write or read without years of instruction. Empathy and appreciating motive are innate (mirror neurons), but understanding that evil is usually the result of ignorance, is not.
Thus, I found it interesting while reading Stigler's History of Statistics, in the section on Euler and Mayer. Euler is probably one of the smartest men to have ever lived, and Wikipedia has a list of many important equations, numbers, and theories named after him. Around 1745 he tried to explain why it appeared that the motion of Jupiter was accelrerating while Saturn was retarding. This was problematic because it seemed to imply Saturn would fly off into the galaxy and Jupiter would crash into the Sun. His solution involved looking at 75 observations of data for 8 parameters in his formula. He basically tried to solve for them exactly, as if the data were all measured perfectly, using substitutions when various parameters would cancel out, but was only able to then pare the set of equations down to six inconsistent linear equations and two unknowns. He found other such solutions with more equations than unknowns, and admitted defeat.
Johann Tabias Mayer, meanwhile, was a cartographer and practical astronomer. In 1750, he took 27 inconsistent equations in thre unkowns and combined them into three equations and three unknowns. Mayer assumed that one could, and should, aggregate the equations,. That is, as a practical astronomer he had previously averaged sets of equations in order to find the 'true' parameters. He correctly intuited that errors canceled out, though he incorrectly thought the error was inversely related to the number of observations, when in fact it is inversely proportional to the square root of the number of observations. Mayer's experimental experience gave him the intuition to believe in the law of large numbers
Euler looked at the problem like a mathematician, and instead intuited that any errors would only compound via aggregation. This makes sense because often mathematicians think that if the error is +/- 2 units, having 4 observations would cause the sum to error by +/- 8 units. Worse!
I think these are really neat because while it's trivial to think that rubes don't understand the odds of winning the lottery or that the odds of a 'head' after 5 'heads' is still 0.5, that's something you can attribute to stupidity. Yet even geniuses don't 'get' certain things that seem like pure logic, but really require a non-intuitive way of looking at things. It's like the story of Paulos Erdos not understanding the Monte Hall problem.